Question

The angle between the line `vecr = (hati + 2hatj + hatk) + lambda(hati + hatj + hatk)` and the plane `vecr * (2hati - hatj + hatk) = 8` is ______.

Options

• `sin^-1 (sqrt(2)/3)`

• `sin^-1 (sqrt(3)/2)`

• `sin^-1 (1/2)`

• `sin^-1 (1/sqrt(2))`

Answer 1

The angle between the line `vecr = (hati + 2hatj + hatk) + lambda(hati + hatj + hatk)` and the plane `vecr * (2hati - hatj + hatk) = 8` is `underlinebb(sin^-1 (sqrt(2)/3)`.

Explanation:

Line:

`vecr = (hati + 2hatj + hatk) + lambda(hati + hatj + hatk)`

So, direction vector of the line is `vecd = (1, 1, 1)`.

Plane:

`vecr * (2hati - hatj + hatk) = 8`

So, normal vector of plane is `vecn = (2, -1, 1)`.

Angle θ between ine and plane is given by:

`sin θ = (|vecd * vecn|)/(|vecd||vecn|)`

Compute dot product:

`vecd * vecn = 1(2) + 1(-1) + 1(1)`

= 2 – 1 + 1

= 2

Magnitudes:

`|vecd| = sqrt(1 + 1 + 1)`

= `sqrt(3)`

`|vecn| = sqrt(4 + 1 + 1)`

= `sqrt(6)`

`sin θ = 2/(sqrt(3) sqrt(6)`

= `2/sqrt(18)`

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

= `sqrt(2)/3`

Therefore, `θ = sin^-1 (sqrt(2)/3)`.