Question

Find the acute angle between the line `bar r = lambda (hat i - hat j + hat k)` and the plane `bar r * (2hat i - hat j + hat k)` = 23.

Answer 1

The acute angle θ between the line `bar r = bar a + lambda bar b` and and the plane `bar r * bar n = d` is given by

sin θ = `|(bar b * bar n)/(|bar b || bar n|)|`       ...(1)

Here, `bar b = hat 2 i + hat 3 j - hat 6 k, bar n = 2 hat i - hat j + hat k`

∴ `bar b * bar n = (hat 2i + hat 3j - hat 6k) * (2 hat i - hat j + hat k)`

= (2)(2) + (3)(– 1) + (– 6)(1)

= 4 – 3 – 6

= – 5

Also, `|bar b| = sqrt (2^2 + 3^2 + (– 6)^2) = sqrt (49)` = 7

`|bar n| = sqrt (2^2 + (– 1)^2 + 1^2) = sqrt (6)`

∴ from (1), we have

sin θ = `|(-5)/(7 sqrt (6))| = 5/(7 sqrt (6))`

∴ θ = `sin^-1 ((5)/(7 sqrt (6)))`.