Question

Consider the following.

Answer 1

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.