vectors and the geometry of spacefac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · the cross...
TRANSCRIPT
![Page 1: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/1.jpg)
VECTORS AND
THE GEOMETRY OF SPACE
12
![Page 2: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/2.jpg)
12.4
The Cross Product
In this section, we will learn about:
Cross products of vectors
and their applications.
VECTORS AND THE GEOMETRY OF SPACE
![Page 3: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/3.jpg)
THE CROSS PRODUCT
The cross product a x b of two
vectors a and b, unlike the dot product,
is a vector.
For this reason, it is also called the vector product.
Note that a x b is defined only when a and b
are three-dimensional (3-D) vectors.
![Page 4: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/4.jpg)
If a = ‹a1, a2, a3› and b = ‹b1, b2, b3›, then
the cross product of a and b is the vector
a x b = ‹a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1›
THE CROSS PRODUCT Definition 1
![Page 5: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/5.jpg)
This may seem like
a strange way of defining
a product.
CROSS PRODUCT
![Page 6: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/6.jpg)
The reason for the particular form of
Definition 1 is that the cross product defined
in this way has many useful properties, as
we will soon see.
In particular, we will show that the vector a x b
is perpendicular to both a and b.
CROSS PRODUCT
![Page 7: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/7.jpg)
In order to make Definition 1 easier
to remember, we use the notation of
determinants.
CROSS PRODUCT
![Page 8: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/8.jpg)
A determinant of order 2 is defined by:
For example,
a bad bc
c d
2 12(4) 1( 6) 14
6 4
DETERMINANT OF ORDER 2
![Page 9: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/9.jpg)
A determinant of order 3 can be defined
in terms of second-order determinants as
follows:
1 2 3
2 3 1 3 1 2
1 2 3 1 2 3
2 3 1 3 1 2
1 2 3
a a ab b b b b b
b b b a a ac c c c c c
c c c
DETERMINANT OF ORDER 3 Equation 2
![Page 10: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/10.jpg)
Observe that:
Each term on the right side of Equation 2 involves
a number ai in the first row of the determinant.
This is multiplied by the second-order determinant
obtained from the left side by deleting the row and
column in which it appears.
DETERMINANT OF ORDER 3
1 2 3
2 3 1 3 1 2
1 2 3 1 2 3
2 3 1 3 1 2
1 2 3
a a ab b b b b b
b b b a a ac c c c c c
c c c
![Page 11: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/11.jpg)
Notice also the minus sign in the second term.
DETERMINANT OF ORDER 3
1 2 3
2 3 1 3 1 2
1 2 3 1 2 3
2 3 1 3 1 2
1 2 3
a a ab b b b b b
b b b a a ac c c c c c
c c c
![Page 12: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/12.jpg)
For example,
1 2 10 1 3 1 3 0
3 0 1 1 2 ( 1)4 2 5 2 5 4
5 4 2
1(0 4) 2(6 5) ( 1)(12 0)
38
DETERMINANT OF ORDER 3
![Page 13: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/13.jpg)
Now, let’s rewrite Definition 1 using
second-order determinants and
the standard basis vectors i, j, and k.
CROSS PRODUCT
![Page 14: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/14.jpg)
We see that the cross product of the vectors
a = a1i +a2j + a3k and b = b1i + b2j + b3k
is:
2 3 1 3 1 2
2 3 1 3 1 2
a a a a a a
b b b b b b a b i j k
CROSS PRODUCT Equation 3
![Page 15: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/15.jpg)
In view of the similarity between Equations 2
and 3, we often write:
1 2 3
1 2 3
a a a
b b b
i j k
a b
CROSS PRODUCT Equation 4
![Page 16: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/16.jpg)
The first row of the symbolic determinant
in Equation 4 consists of vectors.
However, if we expand it as if it were
an ordinary determinant using the rule
in Equation 2, we obtain Equation 3.
CROSS PRODUCT
![Page 17: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/17.jpg)
The symbolic formula in Equation 4 is
probably the easiest way of remembering
and computing cross products.
CROSS PRODUCT
![Page 18: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/18.jpg)
If a = <1, 3, 4> and b = <2, 7, –5>, then
1 3 4
2 7 5
3 4 1 4 1 3
7 5 2 5 2 7
( 15 28) ( 5 8) (7 6)
43 13
i j k
a b
i j k
i j k
i j k
CROSS PRODUCT Example 1
![Page 19: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/19.jpg)
Show that a x a = 0 for any vector a in V3.
If a = <a1, a2, a3>,
then
CROSS PRODUCT Example 2
1 2 3
1 2 3
2 3 3 2 1 3 3 1
1 2 2 1
( ) ( )
( )
0 0 0
a a a
a a a
a a a a a a a a
a a a a
i j k
a a
i j
k
i j k 0
![Page 20: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/20.jpg)
One of the most important
properties of the cross product
is given by the following theorem.
CROSS PRODUCT
![Page 21: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/21.jpg)
The vector a x b is orthogonal
to both a and b.
CROSS PRODUCT Theorem
![Page 22: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/22.jpg)
In order to show that a x b is orthogonal
to a, we compute their dot product as
follows
CROSS PRODUCT Proof
![Page 23: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/23.jpg)
2 3 1 3 1 2
1 2 3
2 3 1 3 1 2
1 2 3 3 2 2 1 3 3 1 3 1 2 2 1
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
( )
( ) ( ) ( )
0
a a a a a aa a a
b b b b b b
a a b a b a a b a b a a b a b
a a b a b a a a b b a a a b a b a a
a b a
CROSS PRODUCT Proof
![Page 24: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/24.jpg)
A similar computation shows that
(a x b) · b = 0
Therefore, a x b is orthogonal to both a and b.
CROSS PRODUCT Proof
![Page 25: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/25.jpg)
Let a and b be represented by directed
line segments with the same initial point,
as shown.
CROSS PRODUCT
![Page 26: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/26.jpg)
Then, Theorem 5 states
that the cross product
a x b points in a direction
perpendicular to the
plane through a and b.
CROSS PRODUCT
![Page 27: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/27.jpg)
It turns out that the direction of a x b
is given by the right-hand rule,
as follows.
CROSS PRODUCT
![Page 28: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/28.jpg)
If the fingers of your right hand curl in
the direction of a rotation (through an angle
less than 180°) from a to b, then your thumb
points in the direction
of a x b.
RIGHT-HAND RULE
![Page 29: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/29.jpg)
We know the direction of the vector a x b.
The remaining thing we need to complete its
geometric description is its length |a x b|.
This is given by the following theorem.
CROSS PRODUCT
![Page 30: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/30.jpg)
If θ is the angle between a and b
(so 0 ≤ θ ≤ π), then
|a x b| = |a||b| sin θ
CROSS PRODUCT Theorem 6
![Page 31: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/31.jpg)
From the definitions of the cross product
and length of a vector, we have:
|a x b|2
= (a2b3 – a3b2)2 + (a3b1 – a1b3)
2 + (a1b2 – a2b1)2
= a22b3
2 – 2a2a3b2b3 + a32b2
2 + a32b1
2
– 2a1a3b1b3 + a12b3
2 + a12b2
2
– 2a1a2b1b2 + a22b1
2
CROSS PRODUCT Proof
![Page 32: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/32.jpg)
= (a12 + a2
2 + a32)(b1
2 + b22 + b3
2)
– (a1b1 + a2b2 + a3b3)2
= |a|2|b|2 – (a . b)2
= |a|2|b|2 – |a|2|b|2 cos2θ [Th. 3 in Sec. 12.3]
= |a|2|b|2 (1 – cos2θ)
= |a|2|b|2 sin2θ
CROSS PRODUCT Proof
![Page 33: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/33.jpg)
Taking square roots and observing that
because sin θ ≥ 0 when
0 ≤ θ ≤ π, we have:
|a x b| = |a||b| sin θ
2sin sin
CROSS PRODUCT Proof
![Page 34: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/34.jpg)
A vector is completely determined by its
magnitude and direction.
Thus, we can now say that a x b is the vector
that is perpendicular to both a and b, whose:
Orientation is determined by the right-hand rule
Length is |a||b| sin θ
CROSS PRODUCT
![Page 35: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/35.jpg)
In fact, that is exactly how
physicists define a x b.
CROSS PRODUCT
![Page 36: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/36.jpg)
Two nonzero vectors a and b are parallel
if and only if
a x b = 0
CROSS PRODUCT Corollary 7
![Page 37: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/37.jpg)
Two nonzero vectors a and b are parallel
if and only if θ = 0 or π.
In either case, sin θ = 0.
So, |a x b| = 0 and, therefore, a x b = 0.
CROSS PRODUCT Proof
![Page 38: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/38.jpg)
The geometric interpretation
of Theorem 6 can be seen from
this figure.
CROSS PRODUCT
![Page 39: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/39.jpg)
If a and b are represented by directed
line segments with the same initial point,
then they determine a parallelogram with
base |a|, altitude |b| sin θ, and
area
A = |a|(|b| sin θ)
= |a x b|
CROSS PRODUCT
![Page 40: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/40.jpg)
Thus, we have the following way of
interpreting the magnitude of a cross
product.
CROSS PRODUCT
![Page 41: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/41.jpg)
The length of the cross product a x b
is equal to the area of the parallelogram
determined by a and b.
CROSS PRODUCT MAGNITUDE
![Page 42: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/42.jpg)
Find a vector perpendicular to the plane
that passes through the points
P(1, 4, 6), Q(-2, 5, -1), R(1, -1, 1)
CROSS PRODUCT Example 3
![Page 43: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/43.jpg)
The vector is perpendicular to
both and .
Therefore, it is perpendicular to the plane
through P, Q, and R.
PQ PR
PQ PR
CROSS PRODUCT Example 3
![Page 44: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/44.jpg)
From Equation 1 in Section 12.2,
we know that:
( 2 1) (5 4) ( 1 6)
3 7
(1 1) ( 1 4) (1 6)
5 5
PQ
PR
i j k
i j k
i j k
j k
CROSS PRODUCT Example 3
![Page 45: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/45.jpg)
We compute the cross product of these
vectors:
3 1 7
0 5 5
( 5 35) (15 0) (15 0)
40 15 15
PQ PR
i j k
i j k
i j k
CROSS PRODUCT Example 3
![Page 46: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/46.jpg)
Therefore, the vector ‹-40, -15, 15›
is perpendicular to the given plane.
Any nonzero scalar multiple of this vector,
such as ‹-8, -3, 3›, is also perpendicular
to the plane.
CROSS PRODUCT Example 3
![Page 47: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/47.jpg)
Find the area of the triangle with vertices
P(1, 4, 6), Q(-2, 5, -1), R(1, -1, 1)
CROSS PRODUCT Example 4
![Page 48: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/48.jpg)
In Example 3, we computed that
The area of the parallelogram with adjacent sides
PQ and PR is the length of this cross product:
2 2 2( 40) ( 15) 15 5 82PQ PR
CROSS PRODUCT Example 4
40, 15,15PQ PR
![Page 49: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/49.jpg)
The area A of the triangle PQR
is half the area of this parallelogram,
that is:
52
82
CROSS PRODUCT Example 4
![Page 50: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/50.jpg)
If we apply Theorems 5 and 6 to the standard
basis vectors i, j, and k using θ = π/2,
we obtain:
i x j = k j x k = i k x i = j
j x i = -k k x j = -i i x k = -j
CROSS PRODUCT
![Page 51: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/51.jpg)
Observe that:
i x j ≠ j x i
Thus, the cross product
is not commutative.
CROSS PRODUCT
![Page 52: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/52.jpg)
Also,
i x (i x j) = i x k = -j
However,
(i x i) x j = 0 x j = 0
So, the associative law for multiplication
does not usually hold.
That is, in general, (a x b) x c ≠ a x (b x c)
CROSS PRODUCT
![Page 53: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/53.jpg)
However, some of the usual
laws of algebra do hold for cross
products.
CROSS PRODUCT
![Page 54: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/54.jpg)
The following theorem
summarizes the properties
of vector products.
CROSS PRODUCT
![Page 55: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/55.jpg)
If a, b, and c are vectors and c is a scalar,
then
1. a x b = –b x a
2. (ca) x b = c(a x b) = a x (cb)
3. a x (b + c) = a x b + a x c
CROSS PRODUCT PROPERTIES Theorem 8
![Page 56: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/56.jpg)
4. (a + b) x c = a x c + b x c
5. a · (b x c) = (a x b) · c
6. a x (b x c) = (a · c)b – (a · b)c
CROSS PRODUCT PROPERTIES Theorem 8
![Page 57: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/57.jpg)
These properties can be proved by writing
the vectors in terms of their components
and using the definition of a cross product.
We give the proof of Property 5 and leave
the remaining proofs as exercises.
CROSS PRODUCT PROPERTIES
![Page 58: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/58.jpg)
Let
a = <a1, a2, a3>
b = <b1, b2, b3>
c = <c1, c2, c3>
CROSS PRODUCT PROPERTY 5 Proof
![Page 59: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/59.jpg)
Then,
a · (b x c) = a1(b2c3 – b3c2) + a2(b3c1 – b1c3)
+ a3(b1c2 – b2c1)
= a1b2c3 – a1b3c2 + a2b3c1 – a2b1c3
+ a3b1c2 – a3b2c1
= (a2b3 – a3b2)c1 + (a3b1 – a1b3)c2
+ (a1b2 – a2b1)c3
=(a x b) · c
CROSS PRODUCT PROPERTY 5 Proof—Equation 9
![Page 60: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/60.jpg)
SCALAR TRIPLE PRODUCT
The product a . (b x c) that occurs
in Property 5 is called the scalar triple
product of the vectors a, b, and c.
![Page 61: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/61.jpg)
Notice from Equation 9 that we
can write the scalar triple product
as a determinant:
1 2 3
1 2 3
1 2 3
( )
a a a
b b b
c c c
a b c
SCALAR TRIPLE PRODUCTS Equation 10
![Page 62: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/62.jpg)
The geometric significance of the scalar
triple product can be seen by considering
the parallelepiped determined by the vectors
a, b, and c.
SCALAR TRIPLE PRODUCTS
![Page 63: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/63.jpg)
The area of the base parallelogram
is:
A = |b x c|
SCALAR TRIPLE PRODUCTS
![Page 64: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/64.jpg)
If θ is the angle between a and b x c,
then the height h of the parallelepiped is:
h = |a||cos θ|
We must use
|cos θ| instead
of cos θ in case
θ > π/2.
SCALAR TRIPLE PRODUCTS
![Page 65: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/65.jpg)
Hence, the volume of the parallelepiped is:
V = Ah
= |b x c||a||cos θ|
= |a · (b x c)|
Thus, we have proved the following formula.
SCALAR TRIPLE PRODUCTS
![Page 66: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/66.jpg)
The volume of the parallelepiped determined
by the vectors a, b, and c is the magnitude of
their scalar triple product:
V = |a ·(b x c)|
SCALAR TRIPLE PRODUCTS Formula 11
![Page 67: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/67.jpg)
If we use Formula 11 and discover that
the volume of the parallelepiped determined
by a, b, and c is 0, then the vectors must lie
in the same plane.
That is, they are coplanar.
COPLANAR VECTORS
![Page 68: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/68.jpg)
Use the scalar triple product to show
that the vectors
a = <1, 4, -7>, b = <2, -1, 4>, c = <0, -9, 18>
are coplanar.
COPLANAR VECTORS Example 5
![Page 69: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/69.jpg)
We use Equation 10 to compute their scalar
triple product:
1 4 7
( ) 2 1 4
0 9 18
1 4 2 4 2 11 4 7
9 18 0 18 0 9
1(18) 4(36) 7( 18) 0
a b c
COPLANAR VECTORS Example 5
![Page 70: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/70.jpg)
Hence, by Formula 11, the volume
of the parallelepiped determined by a, b,
and c is 0.
This means that a, b, and c are coplanar.
COPLANAR VECTORS Example 5
![Page 71: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/71.jpg)
The product a x (b x c) that occurs
in Property 6 is called the vector triple
product of a, b, and c.
Property 6 will be used to derive Kepler’s
First Law of planetary motion in Chapter 13.
Its proof is left as Exercise 46
VECTOR TRIPLE PRODUCT
![Page 72: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/72.jpg)
CROSS PRODUCT IN PHYSICS
The idea of a cross product
occurs often in physics.
![Page 73: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/73.jpg)
CROSS PRODUCT IN PHYSICS
In particular, we consider a force F
acting on a rigid body at a point given
by a position vector r.
![Page 74: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/74.jpg)
For instance, if we tighten a bolt by
applying a force to a wrench, we produce
a turning effect.
CROSS PRODUCT IN PHYSICS
![Page 75: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/75.jpg)
The torque τ (relative to the origin) is defined
to be the cross product of the position and
force vectors
τ = r x F
It measures the
tendency of the body
to rotate about the
origin.
TORQUE
![Page 76: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/76.jpg)
The direction of the torque vector
indicates the axis of rotation.
TORQUE
![Page 77: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/77.jpg)
According to Theorem 6, the magnitude of
the torque vector is
|τ | = |r x F| = |r||F| sin θ
where θ is the angle between the position
and force vectors.
TORQUE
![Page 78: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/78.jpg)
Observe that the only component of F
that can cause a rotation is the one
perpendicular to r—that is, |F| sin θ.
The magnitude of the torque is equal to the area
of the parallelogram determined by r and F.
TORQUE
![Page 79: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/79.jpg)
A bolt is tightened by applying a 40-N force
to a 0.25-m wrench, as shown.
Find the magnitude
of the torque about
the center of the bolt.
TORQUE Example 6
![Page 80: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/80.jpg)
The magnitude of the torque vector is:
|τ| = |r x F|
= |r||F| sin 75°
= (0.25)(40) sin75°
= 10 sin75°
≈ 9.66 N·m
TORQUE Example 6
![Page 81: VECTORS AND THE GEOMETRY OF SPACEfac.ksu.edu.sa/sites/default/files/chap12_sec4.pdf · THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9dc70a9a7f741e2e60be53/html5/thumbnails/81.jpg)
If the bolt is right-threaded, then the torque
vector itself is
τ = |τ| n ≈ 9.66 n
where n is a unit vector directed down
into the slide.
TORQUE Example 6