Question

If vectors `vec a = 3 hati + 2 hatj + λ hatk and vecb = 2 hati - 4 hatj + 5 hatk`, represent the two strips of the Red Cross sign placed outside a doctor’s clinic, then the value of λ is ______.

Options

• 1

• `5/2`

• `2/5`

• 0

Answer 1

If vectors `vec a = 3 hati + 2 hatj + λ hatk and vecb = 2 hati - 4 hatj + 5 hatk`, represent the two strips of the Red Cross sign placed outside a doctor’s clinic, then the value of λ is `bbunderline(2/5)`.

Explanation:

Two vectors `vec a and vec b` are perpendicular if and only if their dot product is zero:

`vec a . vec b = 0`

Given:

`vec a = 3 hati + 2 hatj + λ hatk`

`vecb = 2 hati - 4 hatj + 5 hatk`

The dot product is calculated by multiplying the corresponding components:

(3)(2) + (2)(−4) + (λ)(5) = 0

Solve for λ

6 − 8 + 5λ = 0

−2 + 5λ = 0

5λ = 2

λ = `2/5`