Let V = R3(R) and let v1=[ -1 0 2]^T , v2 = [0 2 -3]^T and v3=[2 2 3]^T.

Question

Let V = R3(R) and let v1=[ -1 0 2]^T , v2 = [0 2 -3]^T and v3=[2 2 3]^T.

Suppose that v4 is another vector which is orthogonal to v1 and v2, and satisfying v2.v4 = -3.

Calculate the following expressions:

(i) v1 . v2,

(ii) v3 . v4,

(iii) (2v1 + 3v2 - v3) . v4,

(iv)||v1|| , ||v2-v3||

(v) d(v1,v2), the distance between v1 and v2,

(vi) the angle between the vectors v1 and v2.

So in the question, it is given that "v4 is another vector which is orthogonal to v1 and v2, and satisfying v2.v4 = -3 ". But, if v4 is orthogonal to v2, shouldn't that mean v2.v4=v4.v2 =0?? But it is given v2.v4= -3

Or am I missing something?

Answer 1

I would say that maybe they meant v4.v3 = -3, but then they ask for v4.v3

So yes, I'd say something is amiss. Maybe they meant only that v4⊥v1.

Answer 2

Thank you!

Answer 3

You are correct that if two vectors are orthogonal, their dot product is equal to zero. In this case, it seems like there might be a mistake in the question, where it states that v2.v4 = -3. It contradicts the statement that v4 is orthogonal to v2.

To clarify, a vector v4 being orthogonal to v2 means that v2 . v4 = 0, not -3. Please double-check the question or provide additional information if available, so we can proceed with the calculations correctly.