Write the position vector of a point dividing the line segment joining points having position vectors $\hat{i} + \hat{j} - 2 \hat{k} and 2 \hat{i} - \hat{j} + 3 \hat{k}$ externally in the ratio 2:3.

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Solution

Let $A$ and $B$ be the points with position vectors $\vec{a} = \hat{i} + \hat{j} - 2 \hat{k}, \vec{b} = 2 \hat{i} - \hat{j} + 3 \hat{k}$ respectively.
Let $C$ divide $A B$ externally in the ratio 2 : 3 such that $A C : C B = 2 : 3$
∴ Position vector of $C$ = $\frac{2 (2 \hat{i} - \hat{j} + 3 \hat{k}) - 3 (\hat{i} + \hat{j} - 2 \hat{k})}{2 - 3}$
= $\frac{4 \hat{i} - 2 \hat{j} + 6 \hat{k} - 3 \hat{i} - 3 \hat{j} + 6 \hat{k}}{- 1}$
= $\frac{\hat{i} - 5 \hat{j} + 12 \hat{k}}{- 1}$
= $- \hat{i} + 5 \hat{j} - 12 \hat{k}$

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Write the position vector of a point dividing the line segment joining points having position vectors i^+j^-2k^ and 2i^-j^+3k^ externally in the ratio 2:3.

Let A and B be the points with position vectors a→=i^+j^-2k^ , b→=2i^-j^+3k^ respectively. Let C divide AB externally in the ratio 2 : 3 such that AC : CB = 2 : 3 ∴ Position vector of C = 22i^-j^+3k^-3i^+j^-2k^2-3 = 4i^-2j^+6k^-3i^-3j^+6k^-1 = i^-5j^+12k^-1 = -i^+5j^-12k^

Write the position vector of a point dividing the line segment joining points having position vectors $\hat{i} + \hat{j} - 2 \hat{k} and 2 \hat{i} - \hat{j} + 3 \hat{k}$ externally in the ratio 2:3.