Parallel and Orthogonal Vectors?

Question

Decompose v into two vectors (vector 1 and vector 2), where vectors 1 is parallel to w and vector 2 is orthogonal to w.

What does the word orthogonal mean here?

Also, what does the question mean by decompose?

(1) v = -3i + 2j, w = 2i + j

(2) v = 1 - 3j, w = 4i - j

Answer 1

"Decompose v into two vectors v1 and v2" means find vectors v1 and v2 such that v = v1 + v2. The vector v1 is parallel to w means that for some scalar k, w = k*v1. The vector v2 is orthogonal to w means that their dot (inner) product is 0. (Orthogonality is the abstract generalization of the concept of perpendicularity.)

Let v1 = ai + bj and v2 = ci + dj. We want
(1) -3i + 2j = (a+c)i + (b + d)j ,
(2) 2i + j = (k*a)i + (k*b)j for some k, and
(3) 2c + d = 0.

From (1) we know that a + c = -3 and b + d = 2. From (2) we know that ka = 2 and kb = 1; this tells us that a = 2b. We can combine this info to get three equations in three unknowns:

2b + c = -3
b + d = 2
2c + d = 0.

The solution is b = -4/5, c = -7/5, and d = 14/5. This tells us that a = -8/5 and k = -5/4 .

The answser to the second part of your question is
v1 = (28/17)i - (7/17)j and v2 = (-11/17)i - (44/17)j .

Answer 2

Orthogonal means "perpendicular," or "at a right angle (at least in two dimensions.

Decompose means find two vectors that add up to the original vector.

* So for (1), vectors parallel to w have the form Ai + (1/2)Aj.
Orthogonal vectors have the form Bi - 2Bj.

So you need A + B = -3, and .5A - 2B = 2

=> B = -3 -A
=> .5A + 6 + 2A = 2
=> 2.5A = -4
=> A = -8/5
and
=> B = -3 + 8/5 = -7/5

Put these back into the equations at *.

Answer 3

If v and w are parallel, then v = kw for any genuine ok. If v and w are orthogonal (i.e. perpendicular), then we tutor that v . w = 0. For the given vectors, v . w does not provide 0 (it is 24 quite) and v can not be expressed as kw. consequently, v and w are neither parallel nor orthogonal.