How Do You Find The Point Of Intersection Of $2$ Vectors?

Question

Two lines $\textbf{v} = \begin{pmatrix} 7 \\ -3 \\ 1 \end{pmatrix} + \begin{pmatrix} -2 \\ 5 \\ 1 \end{pmatrix} t$ and $\textbf{w} = \begin{pmatrix} 8 \\ -1 \\ -1 \end{pmatrix} + \begin{pmatrix} 1 \\ -4 \\ 0 \end{pmatrix} u$ intersect. What point do they intersect at?

Answer 1

We require the $x,y,z$ coordinates to be equal at the point of intersection, so we solve the following set of equations:

$(1)$ $$7-2t = 8+u$$

$(2)$ $$-3+5t = -1-4u$$

$(3)$ $$1+ t = -1$$

Thus from $(3)$ we get $t=-2$

Then from $(1)$ we get $7+4 = 8+u \Rightarrow u = 3$

Then we must check that this satisfies equation $(2)$

$-3+5t = -3 + 5\cdot -2 = -13 $

$-1-4u = -13$ hence there is a point of intsersection

Just plugin $u = 3$ into the equation for $w$ or $t=-2$ into your equation for $v$ to find the point of intersection.

Note the point of intersection you should get is

$\begin{pmatrix} 11\\ -13 \\ -1 \end{pmatrix}$