Problem 224
Find the direction cosines of the line x-y+z-4=0, 2x+y-2z+5=0
Find the direction cosines of the line $x-y+z-4=0,\ 2x+y-2z+5=0$.
Explanation of Method
The direction cosines of a line given as the intersection of two planes.
Step 1: Solution
\[ \vec d=\vec n_1\times\vec n_2=\{1,-1,1\}\times\{2,1,-2\}=\{1,4,3\},\qquad |\vec d|=\sqrt{26}. \]
\[\cos\alpha=\frac{1}{\sqrt{26}},\quad \cos\beta=\frac{4}{\sqrt{26}},\quad \cos\gamma=\frac{3}{\sqrt{26}}\]
Up to a common sign.
Check:
- $\tfrac{1+16+9}{26}=1$;
- $\vec d\cdot\vec n_1=1-4+3=0$;
- $\vec d\cdot\vec n_2=2+4-6=0$.
Result
\[\boxed{\,\cos\alpha=\frac{1}{\sqrt{26}},\quad \cos\beta=\frac{4}{\sqrt{26}},\quad \cos\gamma=\frac{3}{\sqrt{26}}\,}\]