How to find line parallel to direction vector and passing through a specific point?

Question

I am give the point $(1,0,-3)$ and the vector $2i-4j+5k$

Find the equation of the line parallel to vector and passing through point $(1,0,-3)$

Could one use the fact that the dot product between the line and the vector?

Please give me some direction as where to go for this question. I am so lost

Answer 1

$$\frac{x-x_0}{v_x} = \frac{y-y_0}{v_y} = \frac{z-z_0}{v_z}$$ This is line equation in 3D, where $v$ is your vector and $(x_0, y_0, z_0)$ is the given point.

Answer 2

In general, we know that the equation of the line passing through the point $(x_1, y_1, z_1)$ & parallel to the vector $(ai+bj+ck)$ is given as $$\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$$

Hence, the equation of the line passing through the point $(1, 0, -3)$ & parallel to the vector $(2i-4j+5k)$ is given as $$\frac{x-1}{2}=\frac{y-0}{-4}=\frac{z-(-3)}{5}$$ $$\frac{x-1}{2}=\frac{-y}{4}=\frac{z+3}{5}$$ It can also be represented as

$$r(t)=(1, 0, -3)+t(2, -4, 5)$$