Vector parametrization of the line that passes through two vectors.

Question

How would I find the vector parametrization $𝐫(𝑡)$ of the line $L$ that passes through the points $(2,1,4)$ and $(5,6,7)$?

So I found a directional vector: $(3,5,3)$.

What do I do next?

Answer 1

You found a direction vector $(3,5,3)=(5,6,7)-(2,1,4)$,

so the line can be parametrized as $(2,1,4)+t(3,5,3)=(2+3t,1+5t,4+3t)$.

Note that $\mathbf r(0)=(2,1,4)$ and $\mathbf r(1)=(5,6,7)$.