Question

The projection of the line segment joining the points $(1,-1,3)$ and $(2,-4,11)$ on the line joining the points $(-1,2,3)$ and $(3,-2,10)$ is

Answer 1

Use the dot product to find the projection of line segments

Line formed by $(1,-1,3)$ and $(2,-4,11)$ $=(2-1)\hat{i}+(-4+1)\hat{j}+(11-3)\hat{k}\Rightarrow \hat{i}-3\hat{j}+8\hat{k}$

Line formed by $(-1,2,3)$ and $(3,-2,10)$$=(3+1)\hat{i}+(-2-2)\hat{j}+(10-3)\hat{k}\Rightarrow 4\hat{i}-4\hat{j}+7\hat{k}$

Projection of $AB$ on vector $CD$$=\frac{\overrightarrow{AB}·\overrightarrow{CD}}{\left|\overrightarrow{CD}\right|}$

Projection of vector $\hat{i}-3\hat{j}+8\hat{k}$ and $4\hat{i}-4\hat{j}+7\hat{k}$ is $\frac{\left(\hat{i}-3\hat{j}+8\hat{k}\right)\left(4\hat{i}-4\hat{j}+7\hat{k}\right)}{\left|4\hat{i}-4\hat{j}+7\hat{k}\right|}$

$$\begin{gathered} \Rightarrow \frac{4+12+56}{\sqrt{4^2+4^2+7^2}} \\ =\frac{72}{\sqrt{81}} \\ =\frac{72}{9} \\ =8 \end{gathered}$$

Hence, the projection of Projection of $AB$ on vector $CD$ is $8$.