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

• A. $2$
• B. $\sqrt{\frac{271}{53}}$
• C. $\sqrt{\frac{472}{31}}$
• D. $\sqrt{\frac{474}{35}}$

Answer 1

The correct option is D $\sqrt{\frac{474}{35}}$

Let $A(1,0,-1),B(-1,2,2)$
Direction ratios of segment AB are $<2,-2,-3>$.

$\therefore cos\theta =\frac{12\times 1+3(-2)-5(-3)|}{\sqrt{1+9+25}\sqrt{4+4+9}}$

$=\frac{11}{\sqrt{17}\sqrt{35}}=\frac{11}{\sqrt{595}}$

Length of projection $=\left(AB\right)sin\theta$

$=\sqrt{(2{)}^2+(2{)}^2+(3{)}^2}\times \sqrt{1-\frac{121}{595}}$

$=\sqrt{17}\frac{\sqrt{474}}{\sqrt{17}\sqrt{35}}=\sqrt{\frac{474}{35}}$ units