Question

Area of a rectangle having vertices A, B, C, and D with position vectors and respectively is (A) (B) 1 (C) 2 (D)

Answer 1

It is given that the area of a rectangle has vertices A,B,C and D.

The position vectors of vertices are,

OA → =− i ^ + 1 2 j ^ +4 k ^ OB → = i ^ + 1 2 j ^ +4 k ^ OC → = i ^ − 1 2 j ^ +4 k ^ OD → =− i ^ − 1 2 j ^ +4 k ^

The side AB → can be expressed as,

AB → = OB → − OA → =( 1 i ^ + 1 2 j ^ +4 k ^ )−( −1 i ^ + 1 2 j ^ +4 k ^ ) =( 1−( −1 ) ) i ^ +( 1 2 − 1 2 ) j ^ +( 4−4 ) k ^ =2 i ^ +0 j ^ +0 k ^

The side BC → can be expressed as,

BC → = OC → − OB → =( 1 i ^ − 1 2 j ^ +4 k ^ )−( 1 i ^ + 1 2 j ^ +4 k ^ ) =( 1−1 ) i ^ +( − 1 2 − 1 2 ) j ^ +( 4−4 ) k ^ =0 i ^ −1 j ^ +0 k ^

Now, calculate cross product of AB → and BC → .

| AB → × BC → |=| i ^ j ^ k ^ 2 0 0 0 −1 0 | = i ^ ( 0×0−( −1 )×0 )− j ^ ( 2×0−0×0 )+ k ^ ( 2×−1−0×0 ) =0 i ^ −0 j ^ −2 k ^

So, the magnitude of AB → × BC → is,

| AB → × BC → |= ( −2 ) 2 | AB → × BC → |=2

The area of rectangle ABCD is | AB → × BC → |=2.

Therefore, the correct option is option (C).