Question:
The position vector of a point $R$ which divides the line joining $ P(6,\,3,-2) $ and $ Q(3,1,-4) $ Q(3,1, - 4) in the ratio $ 2:1 $ externally is
The position vector of a point $R$ which divides the line joining $ P(6,\,3,-2) $ and $ Q(3,1,-4) $ Q(3,1, - 4) in the ratio $ 2:1 $ externally is
Updated On: Jun 6, 2024
- $ \hat{i}+3\hat{j}-2\hat{k} $
- $ 3\hat{i}-\hat{k} $
- $ -\hat{j}-6\hat{k} $
- $ 2\hat{i}-\hat{j} $
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The Correct Option is C
Solution and Explanation
Let position vector of P and Q are Day
$ OP=6\hat{i}+3\hat{j}-2\hat{k} $ and $ OQ=3\hat{i}+\hat{j}-4\hat{k} $
Now, by section formula
$ OR=\left[ \frac{2\times OQ-1\times OP}{2-1} \right] $
$ =\left( \frac{2(3\hat{i}+\hat{j}-4\hat{k})-(6\hat{i}+3\hat{j}-2\hat{k})}{1} \right) $
$ =\left\{ \frac{6\hat{i}+2\hat{j}-8\hat{k}-6\hat{i}-3\hat{j}+2\hat{k}}{1} \right\} $
$ =(-\hat{j}\,\,-\,6\hat{k}) $
$ OP=6\hat{i}+3\hat{j}-2\hat{k} $ and $ OQ=3\hat{i}+\hat{j}-4\hat{k} $
Now, by section formula
$ OR=\left[ \frac{2\times OQ-1\times OP}{2-1} \right] $
$ =\left( \frac{2(3\hat{i}+\hat{j}-4\hat{k})-(6\hat{i}+3\hat{j}-2\hat{k})}{1} \right) $
$ =\left\{ \frac{6\hat{i}+2\hat{j}-8\hat{k}-6\hat{i}-3\hat{j}+2\hat{k}}{1} \right\} $
$ =(-\hat{j}\,\,-\,6\hat{k}) $
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Concepts Used:
Vector Basics
What is a Vector Quantity?
Vector Quantity is a physical quantity that is specified not only by its magnitude but also by its direction. A vector quantity whose magnitude is equal to one and has direction is called a unit vector.
Examples of vector quantity are-
- Displacement
- Linear momentum
- Momentum
- Acceleration
- Force
- Electric field
- Angular velocity
- Polarization