Question

Find the direction cosines of the vector $\hat{i}+2\hat{j}+3\hat{k}$.

Answer 1

Let $r=i+2j+3k$ be the given vector

The magnitude of vector $r$ is $|r|=\sqrt{1^2+2^2+3^2}=\sqrt{14}$

Let the directional cosines of vector $r$ be $cos\alpha ,cos\beta ,cos\gamma$

We have $cos\alpha =\frac{r.i}{|r|}=\frac{1}{\sqrt{14}}$

We have $cos\beta =\frac{r.j}{|r|}=\frac{2}{\sqrt{14}}$

We have $cos\gamma =\frac{r.k}{|r|}=\frac{3}{\sqrt{14}}$

Therefore the directional cosines of given vector are $\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}}$ and $\frac{3}{\sqrt{14}}$