Let u = 8i + 7j and v = 7i − j. Find the projvu (Projection of u onto v).

Answer 1

There is a formula for proj_vu, the projection of u onto v. It is

proj_vu = [ (u•v) v ] ÷ || v ||²

u•v = 49

(u•v) v = 49 <7, -1> = <343, -49>

||v|| = √50, so ||v||² = 50

proj_vu = [ (u•v) v ] ÷ || v ||² = < (343/50), -(49/50) >

Answer 2

To find proj_vu, we first find the size of u's "shadow" on v (i.e. the scalar projection). This is a simple formula involving the dot product:

^{(u • v)} / _|v|

Here, u • v = (8)(7) + (7)(-1) = 56 - 7 = 49 and |v| = √ [7² + (-1)²] = √50 = 5√2. So our shadow is

^{(u • v)} / _|v| = ⁴⁹ / _5√2

Next, we need to find the unit vector in the direction of v:

^v / _|v| = (¹ / _5√2) (7i - j)

Therefore, our final answer is to multiply the two calculations and simplify:

proj_vu = (⁴⁹ / _5√2)(¹ / _5√2) (7i - j) = 6.86i - 0.98j