What is the projection of $(- i + j + k)$ $(-i + j + k)$ onto $(i - 2 j + 3 k)$ $( i -2j + 3k)$ ?

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Answer 1 · 2017-07-29T06:17:26

No projection since the vectors are perpendicular.

Let $\to b = < - 1, 1, 1 >$ $vecb= <-1,1,1>$ and $\to a = < 1, - 2, 3 >$ $veca= <1,-2,3>$

The vector projection of $\to b$ $vecb$ over $\to a$ $veca$ is

$= \frac{\to a . \to b}{{∣ ∣ ∣ ∣ \to a ∣ ∣ ∣ ∣}^{2}} \cdot \to a$ $=(veca.vecb)/(||veca||^2)*veca$

The dot product is

$\to a . \to b = < - 1, 1, 1 > . < 1, - 2, 3 > = (- 1 \cdot 1) + (1 \cdot - 2) + (1 \cdot 3)$ $veca.vecb=<-1,1,1> . <1,-2,3> =(-1*1)+(1*-2)+(1*3)$

$= - 1 - 2 + 3 = 0$ $=-1-2+3=0$

The vectors $\to a$ $veca$ and $\to b$ $vecb$ are perpendicular.

So there is no projection posiible.

What is the projection of (−i+j+k)(-i + j + k) onto (i−2j+3k) ( i -2j + 3k)?