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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`
Solution
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`
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