Question

Find the vector and the cartesian equations of the line that passes through the points (3, - 2, - 5), (3, - 2, 6).

Answer 1

To find the vector and the cartesian equations of the line that passes through the points (3, - 2, - 5), (3, - 2, 6).

Cartesian form:

$\frac{x - x1}{a} = \frac{y - y1}{b} = \frac{z - z1}{c}$
here point (3, - 2, - 5) through which line passes.

Direction ratio of line are

a= x2-x1 =3-3=0

b= y2-y1 =-2+2=0

c= z2-z1 =6+5=11

So Cartesian form

$\frac{x - 3}{0} = \frac{y + 2}{0} = \frac{z + 5}{11}$
Vector form :

$\vec a+\lambda \vec b$

$\vec a = 3\hat i - 2\hat j - 5\hat k \\ \\ \vec b =0\hat i + 0\hat j + 11 \hat k \\ \\ so \\ \\ (3\hat i - 2\hat j - 5\hat k) + \lambda (11 \hat k)$

Hope it helps you.