Equations of Lines — Question 7

Question 7 – Solution

Strategy Equal inclination means the three direction cosines have equal absolute value. Direction vectors that differ only by an overall nonzero factor define the same line.

All lines Up to reversal, there are four sign patterns. The distinct lines are $$\boxed{\vec r=\left\langle -1,2,0\right\rangle+t\vec d},$$ where $$\vec d\in\{\left\langle 1,1,1\right\rangle,\left\langle 1,1,-1\right\rangle,\left\langle 1,-1,1\right\rangle,\left\langle 1,-1,-1\right\rangle\}.$$ Each direction has direction cosines with absolute value $1/\sqrt 3$.

First-octant test For $\left\langle 1,1,1\right\rangle$, the coordinates are $(-1+t,2+t,t)$, all nonnegative when $t\ge 1$. For $\left\langle 1,-1,1\right\rangle$, they are $(-1+t,2-t,t)$, all nonnegative when $1\le t\le 2$.

For $\left\langle 1,1,-1\right\rangle$ and $\left\langle 1,-1,-1\right\rangle$, the requirements $x\ge 0$ and $z\ge 0$ force incompatible signs of $t$. Thus exactly the first and third listed lines enter the first octant.

Verification The stated intervals satisfy all three coordinate inequalities, including the boundary planes.