Question

What is the projection of `(j+2k)` onto `( i+2j)`?

Answer 1

Scalar projection is `2/sqrt5`

and vector projection is `2/5j+4/5k`

Explanation:

Projection of a vector `veca` onto another vector `vecb` could be scalar projection or vector projection.

A scalar projection is `(veca*vecb)/|veca|` and vector projection is `(veca*vecb)/(veca*veca)*veca`, where `veca` is one on whom we are seeking projection

Here as `vecb=(i+2j)` and `veca=(j+2k)`, hence

`veca*vecb=1xx0+2xx1+0xx2=2`

and `veca*veca=1xx1+2xx2=5` and hence `|veca|=sqrt5`

Hence scalar projection is `2/sqrt5`

and vector projection is `2/5xx(j+2k)=2/5j+4/5k`