Find the position vector (internally) of a point R which divides … - Vidyadip

Question

Find the position vector (internally) of a point R which divides the line joining two points P and Q whose position vectors are $\hat{i}+2 \hat{j}-\hat{k}$ and $-\hat{i}+\hat{j}+\hat{k}$ respectively, in the ratio 2 : 1.

✓

Answer

Given that $\vec{{P}}=\hat{1}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\vec{\mathrm{Q}}=-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$
The $\vec{\mathrm{R}}$ lies on the segment PQ(internal division). If $\mathrm{m}: \mathrm{n}$ is the ratio in which $\vec{\mathrm{R}}$ divides $\mathrm{PQ}$ then
$\vec{\mathrm{R}}=\frac{\mathrm{m} \vec{\mathrm{Q}}+\mathrm{n} \vec{\mathrm{P}}}{\mathrm{m}+\mathrm{n}}$
Given m : n = 2 : 1, m = 2 and n = 1
$\Rightarrow \vec{\mathrm{R}}=\frac{2(-\hat{\imath}+\hat{\jmath}+\hat{\mathrm{k}})+1(\hat{\imath}+2 \hat{\jmath}-\hat{\mathrm{k}})}{2+1}$ = $\frac{-1 \hat{1}+4 \hat{\jmath}+\hat{k}}{3}=\frac{-1}{3} \hat{i}+\frac{4}{3} \hat{\jmath}+\frac{1}{3} \hat{k}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "2 Marks Questions" from Vector Algebra→Vector Algebra — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Differentiate the function with respect to x : $\sin \left( {{x^2} + 5} \right)$

Evaluate the following integrals:$\int\frac{\sec^2\text{x}}{\sqrt{4+\tan^2\text{x}}}\text{ dx}$

Evaluate the following integrals:$\int\frac{\cos\text{x}}{\sqrt{4-\sin^2\text{x}}}\text{ dx}$

Write the smallest equivalence relation on the set A = {1, 2, 3}.

Find the projection of $\vec{\text{a}}$ on $\vec{\text{b}}$ if $\vec{\text{a}}.\vec{\text{b}}=8$ and $\vec{\text{b}}=2\hat{\text{i}}+6\hat{\text{j}}+3\hat{\text{k}}.$

A balloon, which always remains spherical has a variables radius. Find the rate at which its volume is increasing with the radius when the later is $10 cm$.

Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).

Show that the direction cosines of a vector equally inclined to the axes $OX, OY$ and $OZ$ are $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}.$

For the principal values, evaluate the following:
$\text{cosec}^{-1}\Big(2\tan\frac{11\pi}{6}\Big)$

If A is a square matrix of order $3$ such that $|adj\ A| = 64$, find $|A|$.