How can I derive equation of line in 3d?

Question

I started studying 3d geometry and wanted to know how to derive equation of line in 3d from vectors.the equation is r=a+kb.

Answer 1

Have you checked out Paul's Notes? It is a great resource on this subject. I usually don't like to only provide a link and calling it good,, but it's easier to see with the diagrams on the page (and I'm on mobile) http://tutorial.math.lamar.edu/Classes/CalcII/EqnsOfLines.aspx

Answer 2

Let's add some decoration to your equation to distinguish the vectors from the scalar $k$:

$$\vec{r} = \vec{a} + k \vec{b}.$$

So $\vec{a} = (a_x,a_y,a_z)$ is a point in space relative to some origin $(0,0,0)$, and $\vec{b} = (b_x, b_y, b_z)$ is another point in space that defines the direction (relative to the origin) that the line is oriented. The line passes through $\vec{a}$, and the line is parallel to (or coincident with) $\vec{b}$.

Then, the line is all points $\vec{r} = (x,y,z)$ such that the (vector) equation above is satisfied for some $k$.

This can also be expressed as three scalar equations for the individual components:

$$x = a_x + kb_x$$ $$y = a_y + kb_y$$ $$z = a_z + kb_z$$

The parameter $k$ can be anything; all points on the line will be a map from some $k$.