Question

How do you find the scalar and vector projections of b onto a given `a = i + j + k`, `b = i - j + k`?

Answer 1

Saclar projection`1/sqrt3` and Vector projection `1/3(hati+hatj+hatk)`

Explanation:

We have been given two vectors `veca and vec b` , we are to find out the scalar and vector projection of `vec b` onto `vec a`
we have `veca=hati+hatj+hatk` and `vecb=hati-hatj+hatk`
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The scalar projection of `vec b` onto `vec a` means the magnitude of resolved component of `vec b` inthe direction of `vec a` and is given by
The scalar projection of `vec b` onto `vec a` `=(vecb*veca)/|veca|`
`=((hati+hatj+hatk)*(hati-hatj+hatk))/(sqrt(1^2+1^1+1^2))`
`=(1^2-1^2+1^2)/sqrt3=1/sqrt3`

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The Vector projection of `vec b` onto `vec a` means the resolved component of `vec b` in the direction of `vec a` and is given by
The vector projection of `vec b` onto `vec a` `=(vecb*veca)/|veca|^2.(hati+hatj+hatk)`
`=((hati+hatj+hatk)*(hati-hatj+hatk))/((sqrt(1^2+1^1+1^2))^2).(hati+hatj+hatk)`
`=(1^2-1^2+1^2)/3.(hati+hatj+hatk)=1/3(hati+hatj+hatk)`