91Ó°ÊÓ

Given two vectors:

$\stackrel{→}{A}=4\hat{i}-2\hat{j}+4\hat{k}$and $\stackrel{→}{B}=-4\hat{i}+3\hat{k}$.

The unit vector can be found as:

.$\hat{A}=\frac{\stackrel{→}{A}}{\left|A\right|}$

Where,$\stackrel{→}{A}$ is the given vector and$\left|A\right|$ is magnitude of vector.

Vector,$\stackrel{→}{A}=4\hat{i}-2\hat{j}+4\hat{k}$

Magnitude of vector

$$\begin{gathered} \left|A\right|=\sqrt{{\left(4\right)}^2+{\left(-2\right)}^2+{\left(4\right)}^2} \\ =\sqrt{16+4+16} \\ =\sqrt{36} \\ =6 \end{gathered}$$

So the unit vector is:

$$\begin{gathered} \hat{A}=\frac{\stackrel{→}{A}}{\left|A\right|} \\ =\frac{4\hat{i}-2\hat{j}+4\hat{k}}{6} \\ =\frac{2}{3}\hat{i}-\frac{1}{3}\hat{j}+\frac{2}{3}\hat{k} \end{gathered}$$

Unit vector along$\stackrel{→}{A}$ is$\frac{2}{3}\hat{i}-\frac{1}{3}\hat{j}+\frac{2}{3}\hat{k}$

Vector,$\stackrel{→}{B}=-4\hat{i}+3\hat{k}$

Magnitude of vector

$$\begin{gathered} \left|B\right|=\sqrt{{\left(-4\right)}^2+{\left(3\right)}^2} \\ =\sqrt{16+9} \\ =\sqrt{25} \\ =5 \end{gathered}$$

So the unit vector is:

$$\begin{gathered} \hat{B}=\frac{\stackrel{→}{B}}{\left|B\right|} \\ =\frac{-4\hat{i}+3\hat{k}}{5} \\ =\frac{-4}{5}\hat{i}+\frac{3}{5}\hat{k} \end{gathered}$$

Unit vector along$\stackrel{→}{B}$ is$\frac{-4}{5}\hat{i}+\frac{3}{5}\hat{j}$

Sum of unit vectors is

$$\begin{gathered} \hat{A}+\hat{B}=\left(\frac{2}{3}\hat{i}-\frac{1}{3}\hat{j}+\frac{2}{3}\hat{k}\right)+\left(\frac{-4}{5}\hat{i}+\frac{3}{5}\hat{j}\right) \\ =\left(\frac{2}{3}-\frac{4}{5}\right)\hat{i}+\left(-\frac{1}{3}+0\right)\hat{j}+\left(\frac{2}{3}+\frac{3}{5}\right)\hat{k} \\ =\frac{-2}{15}\hat{i}-\frac{1}{3}\hat{j}+\frac{9}{15}\hat{k} \end{gathered}$$

Now, consider a rhombus, whose adjacent sides are unit vectors$\hat{A}$ and $\hat{B}$.

The sum of the vectors (using parallelogram law of vector addition) is given by the diagonal vector.

Diagonal of a rhombus bisects the vertex angle. So, angle between$\hat{A}$ and$\hat{B}$ is bisected by the sum of unit vectors.

Since, $\hat{A}$and$\hat{B}$ are along$\hat{A}$ and $\hat{B}$, so the angle between them is also bisected by the sum of unit vectors.