Question

Write the vector equation of the line $\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}$.

Answer 1

Cartesian equation $=\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}$

Let $\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}=\alpha$

$x=3\alpha +5,y=7\alpha -4,z=2\alpha +6$

Let $(x,y,z)$ be position vectors.

$\therefore x\hat{i}+y\hat{j}+z\hat{k}=5\hat{i}+(-4)\hat{j}+6\hat{k}+\alpha (3\hat{i}+7\hat{j}+2\hat{k})$

vector form : $5\hat{i}+(-4)\hat{j}+6\hat{k}+\alpha (3\hat{i}+7\hat{j}+2\hat{k})$