What is the projection of $< - 3, 1, 3 >$ $< -3 , 1 ,3 >$ onto $< 4, 9, 1 >$ $< 4, 9, 1>$ ?

Question

1494 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-12-08T08:10:20

The vectors are perpendicular, no projection is possible.

The vector projection of $\to v$ $vecv$ onto $\to u$ $vecu$ is

$p r o j_{u} v = \frac{\to u . \to v}{{(∣ ∣ ∣ ∣ \to u ∣ ∣ ∣ ∣)}^{2}} \cdot \to u$ $proj_uv= (vec u. vecv ) /(||vecu||)^2*vecu$

Here,

$\to v = < - 3, 1, 3 >$ $vecv= < -3,1,3>$ and

$\to u = < 4, 9, 1 >$ $vecu = <4,9,1>$

The dot product is

$\to u . \to v = < - 3, 1, 3 > . < 4, 9, 1 >$ $vecu.vecv =<-3,1,3> . <4,9,1>$

$= (4 \cdot - 3) + (9 \cdot 1) + (3 \cdot 1) = - 12 + 9 + 3 = 0$ $=(4*-3)+(9*1)+(3*1) =-12+9+3=0$

As the dot product is $= 0$ $=0$ , the $2$ $2$ vectors $\to u$ $vecu$ and $\to v$ $vecv$ are perpendicular, there is no projection.

What is the projection of < -3 , 1 ,3 ><−3,1,3>< -3 , 1 ,3 > onto < 4, 9, 1><4,9,1>< 4, 9, 1>?