91Ó°ÊÓ

• A vectoris a quantity that contains magnitude along with the direction.
• It's usually represented (drawn) by an arrow with the same direction as the amount and a length proportionate to the magnitude of the quantity.
• A vector does not have only a position, even though it has magnitude and direction.

For two vectors to be equal, they must have the same magnitude and direction.

From the given figure, it is analyzed that$\stackrel{→}{r_1}$ and$\stackrel{→}{r_2}$ are in opposite directions.

Hence, the two given vectors$\stackrel{→}{r_1}$ and $\stackrel{→}{r_2}$are not equal.

Unit vector or $\stackrel{→}{c}$is given by,

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

compute the magnitude of $\stackrel{→}{c}$.

$$\begin{gathered} \left|\stackrel{→}{c}\right|=\sqrt{{\left(400\right)}^2+{\left(200\right)}^2+{(-100)}^2} \\ =\sqrt{16×{10}^4+4×{10}^4+1×{10}^4} \\ =\sqrt{21×{10}^4} \\ =\sqrt{21}×{10}^2 \end{gathered}$$

Substitute$\stackrel{→}{c}$and its magnitude into the formula of unit vector.

$$\begin{gathered} \hat{\mathrm{c}}=\frac{400\hat{\mathrm{i}}+200\mathrm{j}+100\hat{\mathrm{k}}}{\sqrt{21}×{10}^2} \\ =\frac{4}{\sqrt{21}}\hat{\mathrm{i}}+\frac{2}{\sqrt{21}}\hat{\mathrm{j}}-\frac{1}{\sqrt{21}}\hat{\mathrm{k}} \\ =0.873\hat{\mathrm{i}}+0.436\hat{\mathrm{j}}-0.218\hat{\mathrm{k}} \end{gathered}$$

Thus, the unit vector$\hat{c}$ in the direction of$\stackrel{→}{c}$ is $(0.873,0.436,-0.218)$.