Question

Write unit vector in the direction of the sum of vectors:
$\overrightarrow{a}=2\hat{i}-\hat{j}+2\hat{k}$ and $\overrightarrow{b}=-\hat{i}+\hat{j}+3\hat{k}$.

Answer 1

Add the vectors $\overrightarrow{a}=2\hat{i}-\hat{j}+2\hat{k}$ and $\overrightarrow{b}=--\hat{i}+\hat{j}+3\hat{k}$ by writing them in the component form that is

$\overrightarrow{a}=⟨2,-1,2⟩$ and $\overrightarrow{b}=⟨-1,1,3⟩$

Then the sum of the vectors can be computed as follows:

$\overrightarrow{a}+\overrightarrow{b}=⟨2-1,-1+1,2+3⟩$

$=⟨1,0,5⟩$

Therefore, the sum of the vectors is $⟨1,0,5⟩$.

Now the unit vector can be obtained as:

$\Vert \overrightarrow{a}+\overrightarrow{b}\Vert =\sqrt{{\left(1\right)}^2+{\left(0\right)}^2+{\left(5\right)}^2}=\sqrt{1+25}=\sqrt{26}$

Unit Vector: $⟨\frac{1}{\sqrt{26}},0,\frac{5}{\sqrt{26}}⟩$