Question

Find the vector and the cartesian equations of the lines that passes through the origin and (5, - 2, 3).

Answer 1

Solution:

Coordinates of origin A (0,0,0)

Coordinates of point B(5,-2,3)

Direction ratio of the line

5-0,-2-0,3-0

=5,-2,3

standard Cartesian equation of a line passing through a point

$\frac{x - x1}{a} = \frac{y - y1}{b} = \frac{z - z1}{c} \\ \\ \frac{x - 0}{5} = \frac{y - 0}{ - 2} = \frac{z - 0}{3} \\ \\ \frac{x}{5} = \frac{y}{ - 2} = \frac{z}{3}$
Vector equation of a line ,passes through origin and from (5,-2,3)

$\vec a + \lambda \: \vec b \\ \\ = \lambda (5 \: \hat i - 2 \: \hat j + 3 \: \hat k)$
Hope it helps you.