The sine of the angle between the straight line x−$$23=y$$−$$34=z$$−$$45$$ and the plane $$2x$$−$$2y+z=5$$ is

Take (a1$$,$$b1$$,$$c1)=(3$$,$$4$$,$$5) and (a2$$,$$b2$$,$$c2)=(2$$,−$$2$$,$$1). Use $$cos$$$$π$$$$2$$−θ$$=$$|$$a1a2+b1b2+c1c2$$|a $$12+b$$ $$12+c$$ $$12a$$ $$22+b$$ $$22+c$$ $$22$$

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Question

The sine of the angle between the straight line $\frac{x - 2}{3} = \frac{y - 3}{4} = \frac{z - 4}{5}$ and the plane $2 x - 2 y + z = 5$ is

Moderate

Solution

Take $(a_{1}, b_{1}, c_{1}) = (3, 4, 5)$ and $(a_{2}, b_{2}, c_{2}) = (2, - 2, 1)$ .
Use $\cos (\frac{π}{2} - θ) = \frac{| a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} |}{\sqrt{a_{_{1}}^{2} + b_{_{1}}^{2} + c_{_{1}}^{2}} \sqrt{a_{_{2}}^{2} + b_{_{2}}^{2} + c_{_{2}}^{2}}}$

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