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Q.
The ratio in which $\hat{ i }+2 \hat{ j }+3 \hat{ k }$ divides the join of $-2 \hat{ i }+3 \hat{ j }+5 \hat{ k }$ and $7 \hat{ i }-\hat{ k }$ is
Vector Algebra
Solution:
Let the line joining the points with position vectors $-2 \hat{i}+3 \hat{j}+5 \hat{k}$ and $7 \hat{i}-\hat{k}$ be divided in the ratio $\lambda: 1$ by $\hat{i}+2 \hat{j}+3 \hat{k}$
$\therefore \frac{\lambda(7 \hat{ i }-\hat{ k })+(-2 \hat{ i }+3 \hat{ j }+5 \hat{ k })}{\lambda+1}=\hat{ i }+2 \hat{ j }+3 \hat{ k }$
$\Rightarrow(7 \lambda-2) \hat{i}+3 \hat{j}+(5-\lambda) \hat{k}=(\lambda+1) \hat{i}+2(\lambda+1) \hat{j}+3(\lambda+1) \hat{k}$
On equating the coefficient of $\hat{i}$, we get
$7 \lambda-2=\lambda+1 \Rightarrow \lambda=\frac{1}{2}$
Hence, required ratio $=\lambda: 1=1: 2$