First slide

Planes in 3D

Question

The length of projection of the line segment joining the points (1, 0, - 1) and (-1, 2,2) on the plane x+3y-5z=6 is equal to

Moderate

A

2
B

$\sqrt{\frac{271}{53}}$
C

$\sqrt{\frac{472}{31}}$
D

$\sqrt{\frac{474}{35}}$

Solution

Let A (1, 0, -1), B (-1,2,2) Direction ratios of segment AB are < 2, - 2, - 3 >.
$\begin{aligned} ∴ \cos θ = \frac{| 2 \times 1 + 3 (- 2) - 5 (- 3) |}{\sqrt{1 + 9 + 25} \sqrt{4 + 4 + 9}} \\ = \frac{11}{\sqrt{17} \sqrt{35}} = \frac{11}{\sqrt{595}} \end{aligned}$
Length of projection $= (AB) \sin θ$
$\begin{aligned} = \sqrt{(2)^{2} + (2)^{2} + (3)^{2}} \times \sqrt{1 - \frac{121}{595}} \\ = \sqrt{17} \frac{\sqrt{474}}{\sqrt{17} \sqrt{35}} = \sqrt{\frac{474}{35}} units \end{aligned}$

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