The length of the projection of the line segment joining the points −$$1$$,−$$2$$,$$2$$ and $$1$$,$$0$$,$$2on$$ the plane $$x+3y$$−$$5z$$−$$2=0$$ is equal to

Question

The length of the projection of the line segment joining the points  −$$1$$,−$$2$$,$$2$$  and  $$1$$,$$0$$,$$2on$$ the plane  $$x+3y$$−$$5z$$−$$2=0$$ is equal to

Infinity Learn · Accepted Answer

Find the foot of the perpendiculars  P,Q  of the points A−$$1$$,−$$2$$,$$2$$  and  $$B1$$,$$0$$,$$2$$ on the plane  $$x+3y$$−$$5z$$−$$2=0$$ and then find the distance between those two points.The length PQ is projection of   ABon the planeIt implies $$PQ=ABsin$$θ, where θ is angle between the line AB¯ and the normal vector to the plane.Hence, $$PQ=22+221$$−$$cos2$$θ$$=81$$−$$21+23+0$$−$$54+4+01+9+25=81$$−$$8835=8$$×$$1$$−$$835=8$$×$$2735=6635$$                Therefore, the projection is $$6635$$

The length of the projection of the line segment joining the points $(- 1, - 2, 2) and (1, 0, 2)$ on the plane $x + 3 y - 5 z - 2 = 0$ is equal to

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The length of the projection of the line segment joining the points −1,−2,2 and 1,0,2on the plane x+3y−5z−2=0 is equal to

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The length of the projection of the line segment joining the points $(- 1, - 2, 2) and (1, 0, 2)$ on the plane $x + 3 y - 5 z - 2 = 0$ is equal to