Question

The length of the projection of the line segment joining the points (5, 1, 4) and (4, 1, 3) on the plane, x + y + z = 7 is :

Answer 1

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Answer 2

The length of projection is =$\frac{2\sqrt{3} }{3}$

Step-by-step explanation:

Given equation of plane is x+y+z=7

and points are A (5,1,4) and B(4,1,3).

The direction ratio of AB  is (1,0,1) and

Direction ratio of line which is perpendicular to  given plane is (1,1,1).

Let the angle between the lines be ∅.

cos∅=$\frac{1.1+0.1+1.1}{\sqrt{3} \sqrt{2} }$

⇒cos∅=$\frac{2}{\sqrt{6} }$

⇒cos∅=$\sqrt{\frac{2}{3} }$

The length of AB is = $\sqrt{(5-4)^{2}+(1-1)^{2}+(4-3)^{2} }$

                                =$\sqrt{2}$ units

the length of the projection is = |AB|cos∅

                                                  =$\sqrt{2} \sqrt{\frac{2}{3} }$

                                                  =$\frac{2}{\sqrt{3} }$units

                                                   $=\frac{2\sqrt{3} }{3}$units