Question

Three honey bees were found flying along the vectors `vec a = 2 hat i - 3 hat j + hat k, vec b = 4 hat j - 2 hat k and vec c = 3 hat i + 2 hat k` respectively. Find the value of λ such that the path for `vec a + λ vec b` is perpendicular to `vec c`.

Answer 1

Given:

`vec a = 2 hat i - 3 hat j + hat k`

`vec b = 4 hat j - 2 hat k`

`vec c = 3 hat i + 2 hat k`

Finding: `vec a + λ vec b`

`vec a + λ vec b = (2 hat i - 3 hat j + hat k) + λ(4 hat j - 2 hat k)`

`vec a + λ vec b = 2 hat i + (4λ - 3) hat j + (1 - 2 λ) hat k`

Since `(vec a + λ vec b)` is perpendicular to `vec c`, their dot product is zero:

`(vec a + λ vec b) . vec c = 0`

`[2 hat i + (4λ - 3) hat j + (1 - 2 λ) hat k] . (3 hat i + 0 hat j + 2 hat k) = 0`

(2)(3) + (4λ − 3)(0) + (1 − 2λ) (2) = 0

6 + 0 + 2 − 4λ = 0

8 − 4λ = 0

4λ = 8

λ = `8/4`

λ = 2