Question

Assertion $\left(A\right)$ . The direction ratios of the line joining origin and point $(x,y,z)$ must be $x,y,z$

Reason (R): lf $P(x,y,z)$ is a point in space and $|OP|=r,$ then the direction cosines of $OP$ are $\frac{x}{r}$ , $\frac{y}{r}$ , $\frac{z}{r}$

• A. Both A and R are individually true and R is the correct explanation of A
• B. Both A and R individually true but 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 D A is false but R is true

Assertion: Let $P(x,y,z)$

$\overrightarrow{OP}=x\hat{i}+y\hat{j}+z\hat{k}$

DR of $\overrightarrow{OP}$ is any vector parallel to $x\hat{i}+y\hat{j}+z\hat{k}$

So, given assertion is false.

Reason:

$\overrightarrow{OP}=(x\hat{i}+y\hat{j}+z\hat{k})\lambda$

$\overrightarrow{OP}=\lambda \sqrt{x^2+y^2+z^2}=r$

$\alpha ,\beta ,\gamma$ be angles made by $\overrightarrow{OP}$ with $x,y,z$ axes respectively

$\cos \alpha =\frac{\overrightarrow{OP}.\hat{i}}{\left|\overrightarrow{OP}\right|\left|\hat{i}\right|}=\frac{x}{r}$

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

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

DC of $\overrightarrow{OP}$ are $\frac{x}{r},\frac{y}{r},\frac{z}{r}$

Reason is true.