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

Dec 7, 2016

Scalar projection is $\frac{2}{\sqrt{5}}$

and vector projection is $\frac{2}{5} j + \frac{4}{5} k$

#### Explanation:

Projection of a vector $\vec{a}$ onto another vector $\vec{b}$ could be scalar projection or vector projection.

A scalar projection is $\frac{\vec{a} \cdot \vec{b}}{|} \vec{a} |$ and vector projection is $\frac{\vec{a} \cdot \vec{b}}{\vec{a} \cdot \vec{a}} \cdot \vec{a}$, where $\vec{a}$ is one on whom we are seeking projection

Here as $\vec{b} = \left(i + 2 j\right)$ and $\vec{a} = \left(j + 2 k\right)$, hence

$\vec{a} \cdot \vec{b} = 1 \times 0 + 2 \times 1 + 0 \times 2 = 2$

and $\vec{a} \cdot \vec{a} = 1 \times 1 + 2 \times 2 = 5$ and hence $| \vec{a} | = \sqrt{5}$

Hence scalar projection is $\frac{2}{\sqrt{5}}$

and vector projection is $\frac{2}{5} \times \left(j + 2 k\right) = \frac{2}{5} j + \frac{4}{5} k$