53: the scalar product of two vectors © christine crisp “teach a level maths” vol. 2: a2 core...
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53: The Scalar Product 53: The Scalar Product of Two Vectorsof Two Vectors
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 2: A2 Core Vol. 2: A2 Core ModulesModules
The Scalar Product of Two Vectors
Module C4
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The Scalar Product of Two Vectors
In this presentation we are going to look at one of these methods: the scalar product.
Vectors have direction as well as magnitude so it seems strange that we can multiply them together.There are in fact 2 ways of multiplying vectors.
The Scalar Product of Two Vectors
a
bSuppose the angle between two vectors a and b is .
is defined as the angle which is between the vectors when both point towards, or both away from, the point of intersection so, this angle . . .
a
bis NOT . We need to reposition b
The Scalar Product of Two Vectors
a
bSuppose the angle between two vectors a and b is .
is defined as the angle which is between the vectors when both point towards, or both away from, the point of intersection so, this angle . . .
a
bis NOT . We need to reposition b
b
then we see that is an obtuse angle.
The Scalar Product of Two Vectors
cos. abba
a
bSuppose the angle between two vectors a and b is .
The scalar product is written as a . b and is defined as
The dot must NEVER be missed out.
The scalar product is sometimes called the “dot” product.
The result of the scalar product is a scalar quantity not a vector !
The Scalar Product of Two Vectors
Notice that the scalar product uses the magnitudes, a and b, of the vectors as well as the angle between them, so, we get a different answer for:
30cos)10)(8(cos ab
• different size vectors at the same angle,
)2(69 s.f. 30cos)10)(6(cos ab
)2(52 s.f.
cos. abba
8
10
30
630
10
The Scalar Product of Two Vectors
150cos)10)(6(cos ab
• the same size vectors at different angles.
We also have different answers for
30cos)10)(6(cos ab
)2(52 s.f. )2(52 s.f.
630
10
6 150
10
The scalar product was defined so that the answer is unique to any 2 vectors. It will
enable us easily to find the angle between the vectors, even in 3 dimensions.
The Scalar Product of Two Vectors
Perpendicular Vectors
If 0cos ab
either a = 0 or 0cos
a = 0 or b = 0 are trivial cases as they mean the vector doesn’t exist. So, we must have
0cos 90
The vectors are perpendicular.
b = 0 or
The Scalar Product of Two VectorsSUMMAR
Y
cos. abba
The scalar product is written as a . b and is defined as
is defined as the angle which is between the vectors when both point towards, or both away from, the point of intersection.
If and0cos ab ,00 ba and
0cos then and the vectors are perpendicular.
The Scalar Product of Two Vectors
Exercise1. Find the scalar product of the following
pairs of vectors.(a)
660
4(b)
5
3
2. Find the value of a . a for any non-zero vector a.
a
b
a
b (c)
120a
b4
4
3. Find the value of (a) i . i and (b) i . k
The Scalar Product of Two Vectors
1.
(a)
660
4(b)
5
3
a
b
a
b (c)
120a
b4
4
12
0. ba
60cos)6)(4(cos ab(a)
0cos90 (b)
(c) 60 ( not )120 860cos)4)(4(. ba
Solutions:
The Scalar Product of Two Vectors
2. Find the value of a . a for any non-zero vector a.
22 0cos. aaaa
Solution: The angle between a vector and itself is 0
(a) i . i = 1 ( since the magnitude of i is one )
90(b) i . k = 0 ( since )
3. i and k are the unit vectors ( magnitude 1 ) along the x- and z- axes respectively, so
The Scalar Product of Two VectorsThe Scalar Product of 2 Column
Vectors
Then to form , there are quantities to multiply
933 ba .
However, six of these are perpendicular.e.g. is along the x-axis and is along the y-axis.
2 4
a
b
Supposeand
2 34-1
-3 -1
The Scalar Product of Two VectorsThe Scalar Product of 2 Column
Vectors
Supposeand
Then to form , there are quantities to multiply
933 ba .
However, six of these are perpendicular.e.g. is along the x-axis and is along the y-axis.
2 4The scalar product of these components is zero.
a
band
2 34-1
-3 -1
The Scalar Product of Two VectorsThe Scalar Product of 2 Column
Vectors
Supposeand
Then to form , there are quantities to multiply.
933 ba .
The other three multiplications e.g 2 and 3 . . .
However, six of these are perpendicular.e.g. is along the x-axis and is along the y-axis.
2 4The scalar product of these components is zero.
a
band
2 34-1
-3 -1
The Scalar Product of Two VectorsThe Scalar Product of 2 Column
Vectors
Supposeand
Then to form , there are quantities to multiply.
933 ba .
The other three multiplications e.g 2 and 3 . . .
However, six of these are perpendicular.e.g. is along the x-axis and is along the y-axis.
2 4The scalar product of these components is zero.
involve parallel components so the angle between them is zero.
a
band
2 34-1
-3 -1
The Scalar Product of Two VectorsThe Scalar Product of 2 Column
Vectors
Supposeand
a
band
2 34-1
-3 -1
Since ,10cos
The Scalar Product of Two VectorsThe Scalar Product of 2 Column
Vectors
Supposeand
Since ,10cos
a
band
2 34-1
-3 -1
)3)(2(ba .
The Scalar Product of Two VectorsThe Scalar Product of 2 Column
Vectors
Supposeand
10cos
a
band
2 34-1
-3 -1
)4)(1(Since ,10cos
)3)(2(ba .
The Scalar Product of Two VectorsThe Scalar Product of 2 Column
Vectors
Supposeand
10cos
a
band
2 34-1
-3 -1
)4)(1( )1)(3( Since ,10cos
)3)(2(ba .
The Scalar Product of Two VectorsThe Scalar Product of 2 Column
Vectors
Supposeand
10cos
a
band
2 34-1
-3 -1
)4)(1( )1)(3( Since ,10cos
)3)(2(ba .
346
5
The Scalar Product of Two VectorsSUMMAR
Y For the scalar product of 2 column
vectors,
3
2
1
aaa
a
3
2
1
bbb
be.g. and
we multiply the “tops”,and add the results. So,
332211. babababa
“middles”
and “bottoms”
The Scalar Product of Two Vectors
e.g.1 Find the scalar product of the vectors
31
2a
224
band
332211. babababa Solutio
n:
224
31
2. .ba
)2)(3()2)(1()4)(2( 12628
The Scalar Product of Two Vectors
332211. babababa Solution:
143
11
1. .ba
143
e.g.2 Show that the vectors
kjia kjib 43
andare perpendicular.
00cos0. abba
0,0 ba 0cos 90
The Scalar Product of Two Vectors
Exercise1. Find the scalar product of the following
pairs of vectors.
(a)
14
3a
222
band
(b)
23
1a
31
3band
What can you say about the vectors in part (b) ?
The Scalar Product of Two Vectors
(b)
31
3
23
1. .ba
The vectors are perpendicular.
332211. babababa
)2)(1()2)(4()2)(3(. ba 4
)3)(2()1)(3()3)(1( 0
Solutions:
222
14
3. .ba
(a)
The Scalar Product of Two Vectors
The Scalar Product of Two Vectors
The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.
The Scalar Product of Two Vectors
cos. abba
a
bSuppose the angle between two vectors a and b is .
The scalar product is written as a . b and is defined as
The dot must NEVER be missed out.
The scalar product is sometimes called the “dot” product.
The result of the scalar product is a scalar quantity not a vector !
The Scalar Product of Two Vectors
Perpendicular Vectors
If 0cos ab
either a = 0 or 0cos
a = 0 or b = 0 are trivial cases as they mean the vector doesn’t exist. So, we must have
0cos 90
The vectors are perpendicular.
b = 0 or
The Scalar Product of Two VectorsSUMMAR
Y
cos. abba
The scalar product is written as a . b and is defined as
is defined as the angle which is between the vectors when both point towards, or both away from, the point of intersection.
If and0cos ab ,00 ba and
0cos then and the vectors are perpendicular.
The Scalar Product of Two VectorsSUMMAR
Y For the scalar product of 2 column
vectors,
3
2
1
aaa
a
3
2
1
bbb
be.g. and
we multiply the “tops”,and add the results. So,
332211. babababa
“middles”
and “bottoms”
The Scalar Product of Two Vectors
e.g.1 Find the scalar product of the vectors
31
2a
224
band
332211. babababa Solutio
n:
224
31
2. .ba
)2)(3()2)(1()4)(2( 12628
The Scalar Product of Two Vectors
332211. babababa Solution:
143
11
1. .ba
143
e.g.2 Show that the vectors
kjia kjib 43
andare perpendicular.
00cos0. abba
0,0 ba 0cos 90