Meaning of Matrix Multiplication
1. In this problem we will show that multiplication by the matrix
√1
2
√1
2
A=
− √12
√1
2
◦
acts by rotating vectors
v r 45 counterclockwise. As usual, we write the vector v = xi + yj as
x
a column vector
.
y
a) Show that the length of Av is the same as the length of v.
b) Use the dot product to show the angle between v and Av is π/4 radians.
c) Use the cross product to show Av is π/4 radians counterclockwise from v.
Answer: a)
Av =
This has length
required.
√1
2
√1
2
− √12
√1
2
(x − y)2 (x + y)2
+
=
2
2
v
x
y
r
=
x−y
√
2
x+y
√
2
.
x
2 + y 2 . That is, we have shown |Av| = |v| as
b) Using the expression for Av found in part (a) we compute the dot product
�
�
x−y x+y
(x2 + y 2 )
Av · v =
√ ,
√
· �x, y� =
√
.
2
2
2
By part (a) we know |Av| = |v| = x2 + y 2 . So the cosine of the angle between the two
vectors is
Av · v
1
= √ = cos(π/4).
|Av||v|
2
c) We compute the cross product
v × Av =
i
j
k
x2 + y 2
x √
y
√
0
=
√
k.
2
(x − y)/ 2 (x + y)/ 2 0
Since the coefficient of k is positive the right hand rule tells us Av is counterclockwise from
v.
MIT OpenCourseWare
http://ocw.mit.edu
18.02SC Multivariable Calculus
Fall 2010
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