Question

The position vectors of vertices of ΔABC are `4hati - 2hatj; hati + 4hatj - 3hatk` and `-hati + 5hatj + hatk` respectively, then ∠ABC = ______.

Options

• `π/2`

• `π/4`

• `π/6`

• `π/3`

Answer 1

The position vectors of vertices of ΔABC are `4hati - 2hatj; hati + 4hatj - 3hatk` and `-hati + 5hatj + hatk` respectively, then ∠ABC = `underlinebb(π/2)`.

Explanation:

Let position vectors of ΔABC are

A = `4hati - 2hatj`, B = `hati + 4hatj - 3hatk`, C = `-hati + 5hatj + hatk`

Let ∠ABC = θ

`\implies` BA . BC = | BA | | BC | cos θ

`\implies` cos θ = `(BA . BC)/(|BA| |BC|)`

BA = OA – OB

= `(4hati - 2hatj) - (hati + 4hatj - 3hatk)`

= `3hati - 6hatj + 3hatk`

BC = OC – OB

= `(-hati + 5hatj + hatk) - (hati + 4hatj - 3hatk)`

= `-2hati + hatj + 4hatk`

BA . BC = – 6 – 6 + 12 = 0

Now, cos θ = `(BA . BC)/(|BA||BC|)` = 0  ...[∵ BA . BC = 0]

`\implies` θ = `π/2`