Question

Area of a rectangle having vertices A, B, C, and D with position vectors `-hati + 1/2 hatj + 4hatk, hati + 1/2 hatj + 4hatk, and -hati - 1/2j + 4hatk,` respectively is ______.

विकल्प

• `1/2`

• 1

• 2

• 4

Answer 1

Area of a rectangle having vertices A, B, C, and D with position vectors `-hati + 1/2 hatj + 4hatk, hati + 1/2 hatj + 4hatk, and -hati - 1/2j + 4hatk,` respectively is 2.

Explanation:

`vec(OA) = -hati + 1/2hatj + 4hatk,`

` vec(OB) = hati  1/2hatj + 4hatk,`

` vec(OC) = -hati - 1/2hatj + 4hatk`

`vec(AB) = (1 + 1)hati + (1/2 - 1/2)hatj + (4 - 4)hatk `

`= 2hati`

`vec(BC) = (1 - 1)hati + (-1/2 - 1/2)hatj + (4 - 4)hatk`

` = -hatj`

`vec(AB) xx vec(BC) = |(hati, hatj, hatk), (2, 0, 0), (0, -1, 0)| = -2hatk`

`|vec(AB) xx vec(BC)| = sqrt((-2)^2) = 2`