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

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

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Answer 1 · 2016-11-21T14:09:20

The projection is $[\frac{- 1}{3}] [ˆ i - ˆ j + ˆ k]$ $[(-1)/(3)][hati-hatj+hatk]$ .

Explanation:

Let us assume $\to a = (- ˆ i + ˆ j + ˆ k)$ $veca=(-hati+hatj+hatk)$ and $\to b = (ˆ i - ˆ j + ˆ k)$ $vecb=(hati-hatj+hatk)$ .
The projection of vector ' $\to b$ $vecb$ ' on ' $\to a$ $veca$ ' $= ⎡ ⎢ ⎢ ⎣ \frac{\to b . \to a}{{∣ ∣ \to a ∣ ∣}^{2}} ⎤ ⎥ ⎥ ⎦ [\to a]$ $=[(vecb.veca)/(|veca|^2)][veca]$ .
$\to a . \to b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3} = - 1$ $veca.vecb=a_1b_1+a_2b_2+a_3b_3=-1$ .
$∣ ∣ \to a ∣ ∣ = \sqrt{{(- 1)}^{2} + {(1)}^{2} + {(1)}^{2}} = \sqrt{3}$ $|veca|=sqrt[(-1)^2+(1)^2+(1)^2}=sqrt(3)$ .
Substitute the above values in the projection equation we get,
$:.$ projection of $vecb$ on $veca$ is $[(-1)/(sqrt(3))^2][hati-hatj+hatk]=(-1/3)(hati-hatj+hatk)$ .

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

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Explanation:

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