Question

Assertion ($A$): lf the direction ratios of a line are $(-6,2,3)$ then the angles made by the line with coordinate axes are ${\cos }^{-1}\left(\frac{-6}{7}\right),{\cos }^{-1}\left(\frac{2}{7}\right),{\cos }^{-1}\left(\frac{3}{7}\right)$.

Reason $\left(R\right)$: The $d.c$'s of a line $\overline{OP}$ are $\left(\frac{x}{r},\frac{y}{r},\frac{z}{r}\right)$, where $|\overline{OP}|=r$ and $p=(x,y,z)$.

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true and R is not the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Answer 1

The correct option is A Both A and R are true and R is the correct explanation of A

Given $\overrightarrow{r}=-6\hat{i}+2\hat{j}+3\hat{k}=x\hat{i}+y\hat{j}+z\hat{k}$

$\left|\overrightarrow{r}\right|=\sqrt{6^2+2^2+3^2}=7$

Let $\alpha ,\beta ,\gamma$ be angle made with $x,y,z$ axis

$\Rightarrow \cos \alpha =\frac{\overrightarrow{r}.\hat{i}}{\left|\overrightarrow{r}\right|\left|\overrightarrow{i}\right|}=\frac{-6}{7}=\frac{x}{r}$

$\Rightarrow \cos \beta =\frac{\overrightarrow{r}.\hat{j}}{\left|\overrightarrow{r}\right|\left|\overrightarrow{j}\right|}=\frac{2}{7}=\frac{y}{r}$

$\Rightarrow \cos \gamma =\frac{\overrightarrow{r}.\hat{k}}{\left|\overrightarrow{r}\right|\left|\overrightarrow{k}\right|}=\frac{3}{7}=\frac{z}{r}$

Hence, Assertion is true

DC's of $\overrightarrow{OP}$ are $(\frac{x}{r},\frac{y}{r},\frac{z}{r})=(\cos \alpha ,\cos \beta ,\cos \gamma )$

$\therefore$ Assertion and reason are true.