A line passes through A(4,-6,-2) and B(16,-2,4).

Question

A line passes through $A(4,-6,-2)$ and $B(16,-2,4)$. The point $P(a,b,c)$, where $a,b,c$ are non-negative integers, on line $AB$ lies at a distance of $21$ units from $A$. The distance between the points $P(a,b,c)$ and $Q(4,-12,3)$ is equal to:

Answer: 22

Accepted Answer

$\vec{AB}=(12,4,6)$, $|AB|=\sqrt{196}=14$. Unit vector $=\left(\dfrac6{7},\dfrac27,\dfrac37\right)$.
$$P=A+21\left(\frac67,\frac27,\frac37\right)=(4+18,\ -6+6,\ -2+9)=(22,0,7).$$
$$PQ=\sqrt{(22-4)^2+(0+12)^2+(7-3)^2}=\sqrt{324+144+16}=\sqrt{484}=22.$$

Solutions · 1