Problem 248
Examine the mutual position of tfrac…
Examine the mutual position of $\tfrac x{-2}=\tfrac{y-1}{-1}=\tfrac{z-2}1$ and $x+y+3z-7=0$.
Explanation of Method
A line that lies entirely in a given plane.
Step 1: Solution
$\vec n\cdot\vec d=-2-1+3=0$, and $M_0(0,1,2)$ gives $0+1+6-7=0$. So the line lies in the plane. Check with a second point, $(-2,0,3)$: $-2+0+9-7=0$.
\[\text{the line lies in the plane}\]
| $\vec n\cdot\vec d$ | $M_0$ in plane | Position |
|---|---|---|
| $\ne0$ | – | one piercing point |
| $0$ | no | parallel (Problem 247) |
| $0$ | yes | in the plane (this problem) |
$M_0$
Result
\[\boxed{\,\text{the line lies in the plane}\,}\]