Question

The sine of the angle between the straight line `(x - 2)/3 = (y - 3)/4 = (z - 4)/5` and the plane 2x – 2y + z = 5 is ______.

Options

• `10/(6sqrt(5))`

• `4/(5sqrt(2))`

• `(2sqrt(3))/5`

• `sqrt(2)/10`

Answer 1

The sine of the angle between the straight line `(x - 2)/3 = (y - 3)/4 = (z - 4)/5` and the plane 2x – 2y + z = 5 is `sqrt(2)/10`.

Explanation:

Given that, l = `(x - 2)/3 = (y - 3)/4 = (z - 4)/5`

And P: 2x – 2y + z = 5

D’ratios of the line are 3, 4, 5

And d’ratios of the normal to the plane are 2, – 2, 1

∴ `sin theta = (3(2) + 4(-2) + 5(1))/(sqrt(9 + 16 + 25) * sqrt(4 + 4 + 1))`

⇒ `sin theta = (6 - 8 + 5)/(sqrt(50).3)`

⇒ `3/(5sqrt(2).3)`

= `1/(5sqrt(2)`

= `sqrt(2)/10`