Equations of Lines — Question 3

Question 3 – Solution

Strategy Parameterize the segment from $A$ to $B$. The known $x$-coordinate of $P$ determines the parameter, after which the other coordinates and division ratio follow.

Parameter and coordinates Since $B-A=\left\langle 4,6,-4\right\rangle$, $$\vec r(t)=\left\langle 1,-2,3\right\rangle+t\left\langle 4,6,-4\right\rangle.$$ At $P$, $1+4t=4$, so $t=3/4$. Therefore $$u=-2+6\left(\tfrac 34\right)=\tfrac 52,\qquad v=3-4\left(\tfrac 34\right)=0,$$ and $$\boxed{P=\left(4,\tfrac 52,0\right)}.$$

Division ratio The parameter places $P$ three-fourths of the way from $A$ to $B$, so $$\boxed{AP:PB=3:1}.$$

Verification We have $P-A=\tfrac 34(B-A)$ and $B-P=\tfrac 14(B-A)$; taking lengths preserves the ratio $3:1$.