"Points $A$ and $B$ have coordinates $(4,1)$ and $(2,-5)$ respectively. Find a vector equation for the line which passes through the point $A$, and which is perpendicular to the line $AB$."
Any help in this would be greatly appreciated with the steps outlined on how you would work it out as I'm having a hard time understanding vector equations.
Thanks!
Notice that a vector in the direction of line $AB$ is $\vec{v} = (2-4, -5-1) = (-2,-6) = -2 (1,3) $. Now, a perpendicular vector to $\vec{v}$ is for example $\vec{w} = (3,-1)$ since
$$ \vec{v} \cdot \vec{w} = 1 \cdot 3 - 3 \cdot 1 = 0 $$
Therefore, the line you want have direction vector $\vec{w} = (3,-1)$ and passes through the point $(4,1)$. Can you finish it?