Question

Find the position vector of point \( C \) which divides the line segment joining points \( A \) and \( B \) having position vectors \( \vec{a} = \hat{i} + 2\hat{j} - \hat{k} \) and \( \vec{b} = -\hat{i} + \hat{j} + \hat{k} \), respectively, in the ratio 4:1 externally. Further, find \( |\vec{AB}| : |\vec{BC}| \).

Hint:

To find the ratio of line segments, calculate the magnitudes of the respective vectors and simplify.

Answer 1

Step 1: Determine the position vector of point \( C \). The position vector of \( C \), which divides \( AB \) externally in the ratio \( 4:1 \), is calculated as: \[ \vec{r} = \frac{4\vec{b} - \vec{a}}{3}. \] Substituting the given vectors \( \vec{a} = \hat{i} + 2\hat{j} - \hat{k} \) and \( \vec{b} = -\hat{i} + \hat{j} + \hat{k} \): \[ \vec{r} = \frac{4(-\hat{i} + \hat{j} + \hat{k}) - (\hat{i} + 2\hat{j} - \hat{k})}{3}. \] After simplification: \[ \vec{r} = \frac{-4\hat{i} + 4\hat{j} + 4\hat{k} - \hat{i} - 2\hat{j} + \hat{k}}{3}. \] Combining like terms yields: \[ \vec{r} = \frac{-5\hat{i} + 2\hat{j} + 5\hat{k}}{3}. \] Step 2: Calculate the magnitude of \( \vec{AB} \). The vector \( \vec{AB} \) is found by: \[ \vec{AB} = \vec{b} - \vec{a} = (-\hat{i} + \hat{j} + \hat{k}) - (\hat{i} + 2\hat{j} - \hat{k}) = -2\hat{i} - \hat{j} + 2\hat{k}. \] Its magnitude is: \[ |\vec{AB}| = \sqrt{(-2)^2 + (-1)^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3. \] Step 3: Calculate the magnitude of \( \vec{BC} \). The vector \( \vec{BC} \) is determined by: \[ \vec{BC} = \vec{b} - \vec{r} = (-\hat{i} + \hat{j} + \hat{k}) - \frac{-5\hat{i} + 2\hat{j} + 5\hat{k}}{3}. \] Simplifying this expression gives: \[ \vec{BC} = \frac{2\hat{i} - \hat{j} - 2\hat{k}}{3}. \] The magnitude of \( \vec{BC} \) is: \[ |\vec{BC}| = \sqrt{\left(\frac{2}{3}\right)^2 + \left(\frac{-1}{3}\right)^2 + \left(\frac{-2}{3}\right)^2} = \sqrt{\frac{4}{9} + \frac{1}{9} + \frac{4}{9}} = \sqrt{\frac{9}{9}} = 1. \] Step 4: Determine the ratio \( |\vec{AB}| : |\vec{BC}| \). The ratio of the magnitudes is: \[ |\vec{AB}| : |\vec{BC}| = 3 : 1. \] Step 5: Present the final results. The position vector of \( C \) is: \[ \vec{r} = \frac{-5\hat{i} + 2\hat{j} + 5\hat{k}}{3}. \] The ratio \( |\vec{AB}| : |\vec{BC}| \) is \( 3:1 \).