Question

Find the angle between the line `bar r = (hat i + 2hat j + hat k) + lambda(hat i + hat j + hat k)` and the plane `bar r *(2hat i + hat j + hat k) = 8`.

Answer 1

Given line is `bar r = (hat i + 2hat j + hat k) + lambda(hat i + hat j + hat k)`

Given plane is `bar r *(2hat i + hat j + hat k) = 8`

Let `bar b = hat i + hat j + hat k` and

`bar n = 2 hat i + hat j + hat k`

`|bar b| = sqrt(1 + 1 + 1)`

= `sqrt 3` and 

`|bar n| = sqrt (4 + 1 + 1)`

= `sqrt 6`

If θ is the acute angle between the given line and plane, then

`sin theta = |(bar b * bar n)/(|bar b| * |bar n|)|`    ...(i)

We have

`bar b * bar n = (hat i + hat j + hat k) * (2 hat i + hat j + hat k)`

= (1) (2) + (1) (1) + (1) (1)

= 2 + 1 + 1

= 4

Also, `|bar b| = sqrt3`

`|bar n| = sqrt 6`

Put in equation (i), we get

`therefore sin theta = |4/(sqrt 3 * sqrt 6)|`

= `|4/(3 sqrt 2)|`

= `(2 sqrt 2)/3`

∴ `theta = sin^-1 ((2 sqrt2)/3)`