Answer
< 1, -8, -10 >
Work Step by Step
a = <4, 3, -2>
b = <2, -1, 1>
First find the cross product of vector a and vector b:
a x b = | i j k | = | 3 -2| | 4 -2| | 4 3|
| 4 3 -2| |-1 1 | i - | 2 1 | j + |2 -1| k
| 2 -1 1 |
= (3 - 2) i - (4 - (-4)) j + (-4 - 6) k
= i - 8j - 10 k
= < 1, -8, -10>
To prove a x b orthogonal to a, find a dot ( a x b):
a⋅(a x b)
= < 4, 3, -2> ⋅<1, -8, -10>
= 4 + -24 + 20
= 0
Since the dot product is 0, the cross product is orthogonal to vector a because cos (theta) = 0 implies a theta of 90 degrees, thus orthogonal.
To prove a x b orthogonal to b, find b dot ( a x b):
b⋅(a x b)
= < 2, -1, 1> ⋅<1, -8, -10>
= 2 + 8 + -10
= 0
Since the dot product is 0, the cross product is orthogonal to vector a because cos (theta) = 0 implies a theta of 90 degrees, thus orthogonal.