Ex 10.4, 12 - Area of a rectangle having vertices A, B, C, D

Ex 10.4, 12 Area of a rectangle having vertices A, B, C and D with position vectors − 𝑖﷯ + 1﷮2﷯ 𝑗﷯ + 4 𝑘﷯, 𝑖﷯ + 1﷮2﷯ 𝑗﷯ + 4 𝑘﷯, 𝑖﷯ − 1﷮2﷯ 𝑗﷯ + 4 𝑘﷯, − 𝑖﷯ − 1﷮2﷯ 𝑗﷯ + 4 𝑘﷯ respectively is (A) 1﷮2﷯ (B) 1 (C) 2 (D) 4 𝑂𝐴﷯ = − 𝑖﷯ + 1﷮2﷯ 𝑗﷯ + 4 𝑘﷯ = −1 𝑖﷯ + 1﷮2﷯ 𝑗﷯ + 4 𝑘﷯ 𝑂𝐵﷯ = 𝑖﷯ + 1﷮2﷯ 𝑗﷯ + 4 𝑘﷯ = 1 𝑖﷯ + 1﷮2﷯ 𝑗﷯ + 4 𝑘﷯ 𝑂𝐶﷯ = 𝑖﷯ − 1﷮2﷯ 𝑗﷯ + 4 𝑘﷯ = 1 𝑖﷯ - 1﷮2﷯ 𝑗﷯ + 4 𝑘﷯ 𝑂𝐷﷯ = − 𝑖﷯ − 1﷮2﷯ 𝑗﷯ + 4 𝑘﷯ = −1 𝑖﷯ – 1﷮2﷯ 𝑗﷯ + 4 𝑘﷯ Since rectangle is also a parallelogram Area of rectangle ABCD = 𝐴𝐵﷯× 𝐵𝐶﷯﷯ Now, Magnitude of 𝐴𝐵﷯× 𝐵𝐶﷯ = ﷮02+02+ −2﷯2﷯ 𝑨𝑩﷯× 𝑩𝑪﷯﷯ = ﷮4﷯ = 2 Therefore, area of rectangle ABCD = 𝐴𝐵﷯× 𝐵𝐶﷯﷯ = 2 Hence, (C) is the correct option