3D Vector Equation

Question

Consider the points $A (1, 5, 4)$, $B (3, 1, 2)$ and $D (3, k, 2)$, with $\overline{AD}$ being perpendicular to $\overline{AB}$.

(i) Find $AB$

(ii) $AD$ , give the answer terms of $k$.

Show that $k = 7$.

Answer 1

I would say

$\vec{AB}= (3, 1, 2)-(1, 5, 4)=(2,-4,-2);\,\,\,\vec{AD}=(3, k, 2)-(1, 5, 4)=(2,k-5,-2)$

$\vec{AB}$ is perpendicular to $\vec{AD}$ when $\vec{AB}\cdot \vec{AD}=0 \Rightarrow$

$(2,-4,-2)\cdot (2,k-5,-2)=2\cdot 2+ (-4)\cdot(k-5)+ (-2)\cdot (-2)=0 \Rightarrow k = 7$