Find a) the angle between u and v, b) projv u and c) scalv u. Recall that a ? b = |a||b| cos ?.

Question

Find a) the angle between u and v, b) projv u and c) scalv u. Recall that a ? b = |a||b| cos θ.

 

 

u=-3i+4k

v=-4i+j+5k

Answer 1

|u| = 5 and |v| = √42, while u·v = 12 + 0 + 20 = 32.

 

So,

 

cos θ = 32/(5√42), and

 

θ = cos^-1 (32/(5√42)) ≈ 0.1580 rad.

 

Next, proj_v u =(u·v)/(v·v) v = 32/42 (-4i + j + 5k) = 16/21 (-4i + j + 5k) = -64/21 i + 16/21 j + 80/21 k .

 

I assume scal_v u is the scalar projection of u onto v, which is |u| cos θ = 5 · 32/(5√42) = 32/√42 . This happens to be the magnitude of proj_v u, since it points in the same general direction as v (u·v is positive).

 

If my assumption was wrong, you may need to correct my answer to the last question.