Math 210, Quiz 1 (9/7/04)
(1) Find the scalar and vector projections of vector < −1, 2, 3 > onto
vector < 3, 1, 0 >.
Answer: Let a =< −1, 2, 3 > and b =< 3, 1, 0 >. We like to find the scalar
and vector projections from a onto b. (The answer is different for projection
from b onto a. This is a common mistake.)
Scalar projection from a onto b is:
|a| cos θ =
√
a·b
= −1/ 10 .
|b|
Vector projection from a onto b is:
(
a·b b
1 b
1
)
= −√
= − < 3, 1, 0 > .
|b| |b|
10
10 |b|
(2) Given coordinates of P , Q, R be (0,1,2), (2,4,5), (-1,0,1), respectively.
Find a vector that is orthogonal to the plane formed by P, Q, R.
Answer: P~Q =< 2, 4, 5 > − < 0, 1, 2 >=< 2, 3, 3 > and P~R =< −1, 0, 1 >
− < 0, 1, 2 >=< −1, −1, −1 >. Now,
P~Q × P~R =< 0, −1, 1 > .
This vector is orthogonal to the plane formed by P, Q, R.
1