Question

The Cartesian equation of a line is $\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}$, write its vector form.

Answer 1

The given Cartesian equation is $\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}$
The above line passes through the point (5, - 4, 6). The position vector of this point is $a=5\hat{i}-4\hat{j}+6\hat{k}$
Also, the direction ratios of the given line are 3, 7 and 2 . This means that the line is in the direction of vector, $b=3\hat{i}+7\hat{j}+2\hat{k}$
It is known that the line through position vector a and in the direction of the vector b is given by the equation, $r=a+\lambda b,\lambda ϵR$
$\therefore r=5\hat{i}-4\hat{j}+6\hat{k}+\lambda (3\hat{i}+7\hat{j}+2\hat{k})$
This is the required equation of the given line in vector form.