the cross product
DESCRIPTION
The Cross Product. Third Type of Multiplying Vectors. Cross Products. Determinants. It is much easier to do this using determinants because we do not have to memorize a formula. Determinants were used last year when doing matrices - PowerPoint PPT PresentationTRANSCRIPT
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The Cross ProductThe Cross ProductThe Cross ProductThe Cross Product
Third Type of Multiplying VectorsThird Type of Multiplying Vectors
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Cross Products
1 1 1 2 2 2
1 2 2 1 1 2 2 1 1 2 2 1
If v and
are two vectors in space, the cross product
v is defined as the vector
v ( ) ( ) ( )
a i b j c k w a i b j c k
w
w b c b c i a c a c j a b a b k
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Determinants• It is much easier to do this using
determinants because we do not have to memorize a formula.
• Determinants were used last year when doing matrices
• Remember that you multiply each number across and subtract their
products
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Finding Cross Products Using Equation
2 3
3 2 Find v
( 3 1 2 1 (2 ( 1) 3 1) (2 ( 2) ( 3 3)
(3 2) ( 2 3) ( 4 9)
5 5 5
v i j k
w i j k w
i j k
i j k
i j k
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Evaluating a Determinant
2 32 2 1 3 4 3 1
1 2
6 5(6 1) 5 2 6 10 4
2 1
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Evaluating Determinants
1 4 2 4 2 12 1 4
3 1 1 1 1 31 3 1
(1 1 3 4) 2 1 1 4 2 3 1 1
11 2 5
A B C
A B C
A B C
A B C
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Using Determinants to Find Cross Products
• This concept can help us find cross products.
• Ignore the numbers included in the column under the vector that will be inserted when setting up the determinant.
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Using Determinants to Find Cross Products
• Find v x w given• v = i + j• w = 2i + j + k
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Using Determinants to Find Cross Products
1 0 1 0 1 11 1 0
1 1 2 1 2 12 1 1
(1 0) (1 0) (1 2)
i j k
i j k
i j k
i j k
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Using Determinants to Find Cross Products
• If v = 2i + 3j + 5k and w = i + 2j + 3k,
• find • (a) v x w• (b) w x v• (c) v x v
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Using Determinants to Find Cross Products
( )
2 3 5
1 2 3
3 5 2 5 2 3
2 3 1 3 1 2
(9 10) (6 5) (4 3)
a
i j k
v w
i j k
i j k
i j k
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Using Determinants to Find Cross Products
( )
1 2 3
2 3 5
2 3 1 3 1 2
3 5 2 5 2 3
(10 9) (5 6) 3 4
Notice that this is the same values as but
the signs are opposite. (Neg. is now pos. and vice
versa)
b
i j k
w v
i j k
i j k
i j k
v w
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Using Determinants to Find Cross Products
2 3 5
2 3 5
3 5 2 5 2 3
3 5 2 5 2 3
(15 15) (10 10) (6 6)
0
This leads us to one of the properties of cross
products.
i j k
v v
i j k
i j k
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Algebraic Properties of the Cross Product
• If u, v, and w are vectors in space and if is a scalar, then
• u x u = 0• u x v = -(v x u)• (u x v) = (u) x v = u x (v)• u x (v + w) = (u x v) + (u x w)
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Examples• Given u = 2i – 3j + k v = -3i + 3j
+ 2k• w = i + j + 3k• Find• (a) (3u) x v• (b) v . (u x w)
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Examples(3 ) 3(2 3 ) 6 9 3
(3 ) 6 9 3
3 3 2
9 3 6 3 6 9
3 2 3 2 3 3
( 18 9) (12 ( 9) (18 27)
27 21 9
u i j k i j k
i j k
u v
i j k
i j k
i j k
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Examples
2 3 1
1 1 3
3 1 2 1 2 3
1 3 1 3 1 1
( 9 1) (6 1) (2 ( 3))
10 5 5
( 3 ( 10)) (3 ( 5)) (2 5)
30 15 10 25
i j k
u w
i j k
i j k
i j k
v u w
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Geometric Properties of the Cross Product
• Let u and v be vectors in space• u x v is orthogonal to both u and v.• ||u x v|| = ||u|| ||v|| sin where is
the angle between u and v.• ||u x v|| is the area of the
parallelogram having u ≠ 0 and v ≠ 0 as adjacent sides
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Geometric Properties of the Cross Product
• u x v = 0 if and only if u and v are parallel.
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Finding a Vector Orthogonal to Two Given
Vectors• Find a vector that is orthogonal to • u = 2i – 3j + k and v = i + j + 3k
• According to the preceding slide, u x v is orthogonal to both u and v. So to find the vector just do u x v
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Finding a Vector Orthogonal to Two Given
Vectors
2 3 1
1 1 3
3 1 2 1 2 3
1 3 1 3 1 1
( 9 1) (6 1) (2 ( 3))
10 5 5
i j k
u v
i j k
i j k
i j k
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Finding a Vector Orthogonal to Two Given
Vectors • To check to see if the answer is
correct, do a dot product with one of the given vectors. Remember, if the dot product = 0 the vectors are orthogonal
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Finding a Vector Orthogonal to Two Given
Vectors
(2 10) ( 3 5) (1 5)
0
u u v
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Finding the Area of a Parallelogram
• Find the area of the parallelogram whose vertices are P1 = (0, 0, 0),
• P2 = (3,-2, 1), P3 = (-1, 3, -1) and • P4 = (2, 1, 0)
• Two adjacent sides of this parallelogram are u = P1P2 and v = P1P3.
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Finding the Area of the Parallelogram
1 2
1 3
1 2 1 3
1 2 1 3
2 2 2
3 0, 2 0,1 0 3 2
( 1 0,3 0, 1 0) 3
3 2 1 2 7
1 3 1
Area of a Parallelogram =
1 2 7 1 4 49 54 3 6
PP i j k
PP i j k
i j k
PP PP i j k
PP PP
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Your Turn• Try to do page 653 problems 1 –
47 odd.