Question

How do you multiply unit vectors $?$

Answer 1

To multiply unit vectors:

Let consider three mutually perpendicular axes.

These are $X$, $Y$ and $Z$. Suppose unit vectors along those three axes are$\hat{i}$, $\hat{j}$ and $\hat{k}$.

Let us find their scalar product.

$\hat{i}.\hat{i}=\left|\hat{i}\right|.\left|\hat{i}\right|Cos0°$; as angle between two unit vectors is $0°$ $\left[\mathbf{\because }\mathit{a}\mathbf{·}\mathit{b}\mathbf{}\mathbf{=}\mathbf{}\left|a\right|\left|b\right|\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{\theta }\right]$

$$\begin{gathered} =1\times 1\times 1 \\ =1 \end{gathered}$$

similarly, $\hat{j}.\hat{j}=1$ and $\hat{k}.\hat{k}=1$

so, $\hat{i}.\hat{i}=\hat{j}.\hat{j}=\hat{k}.\hat{k}=1$

Now,

$\hat{i}.\hat{j}=\left|\hat{i}\right|\left|\hat{j}\right|\cos 90°$ (angle between $X$ and $Y$ is $90°$) $\left[\mathbf{\because }\mathit{a}\mathbf{·}\mathit{b}\mathbf{}\mathbf{=}\mathbf{}\left|a\right|\left|b\right|\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{\theta }\right]$

$$\begin{gathered} =1\times 1\times 0 \\ =0 \end{gathered}$$

Similarly,

$\hat{j}.\hat{k}=0$and $\hat{k}.\hat{i}=0$

then, $\hat{i}.\hat{j}=\hat{j}.\hat{k}=\hat{k}.\hat{i}=0$

Hence, By this method you can multiply unit vectors.