Answer: 1. (a) (23/13)√26 (b) <46/67, 207/67, 161/67>
2. (a) (29/62)√62 (b) (145/41)j + (161/41)k
Step-by-step explanation:
The Projection of u onto v is given as
u.v/|u|
and the Vector Component of u orthogonal to v is
(u.v/|v|²)v
We will need to find
* The dot products of u and v, u.v
* The magnitude of u, |u|
* The magnitude of v, |v|
And we are good to go.
Now, let's do that for number 1.
u.v = 4.3 + 3.9 + 1.7
= 12 + 27 + 7
= 46 (Note that dot (.) here is a product, not decimal point)
|u| = √(4² + 3² + 1²)
= √(16 + 9 + 1)
= √26
|v| = √(2² + 9² + 7²)
= √(4 + 81 + 49)
= √134
Now we can find (a) and (b)
(a) using u.v/|u|, we have
Proj = 46/√26
= (23/13)√26
(b) using (u.v/|v|²)v, we have
Vector Component = (46/136)<2, 9, 7>
= <46/67, 207/67, 161/67>
Similarly for number 2.
u.v = 1.5 + 6.4
= 5 + 24
= 29
|u| = √(5² + 1² + 6²)
= √(25 + 1 + 36)
= √62
|v| = √(5² + 4²)
= √(25 + 16)
= √41
(a) Proj = 29/√62
(b) Vector Component = (29/41)(5j + 4k)
= (145/41)j + (116/41)k
And we are done.
