Supplementary Exercise Q3

Transcript

Supplementary Exercise Q3 Find the volumes of the following parallelepipeds whose three co –terminus edges are (i) 𝑎﷯ = 2 𝑖﷯ − 3 𝑗﷯ + 4 𝑘﷯, 𝑏﷯ = 3 𝑖﷯ − 𝑗﷯ + 2 𝑘﷯, and 𝑐﷯ = 𝑖﷯ + 2 𝑗﷯ − 𝑘﷯, Given, 𝑎﷯ = 2 𝑖﷯ − 3 𝑗﷯ + 4 𝑘﷯ , 𝑏﷯ = 3 𝑖﷯ – 𝑗﷯ + 2 𝑘﷯ , 𝑐﷯ = 𝑖﷯ + 2 𝑗﷯ – 𝑘﷯ Volume of parallelepiped = 𝒂﷯ 𝒃﷯ 𝒄﷯﷯ = 2﷮−3﷮4﷮3﷮−1﷮2﷮1﷮2﷮−1﷯﷯ = 2 −1×−1﷯−(2×2) ﷯ − (−3) 3×−1﷯−(1×2) ﷯ + 4 3×2﷯−(1×−1)﷯ = 2 1−4﷯+3 −3−2﷯+4 6+1﷯ = 2(–3) + 3 (–5) + 4(7) = –6 – 15 + 28 = 7 Supplementary Exercise Q3 Find the volumes of the following parallelepipeds whose three co –terminus edges are (ii) 𝑎﷯ = 𝑖﷯ − 2 𝑗﷯ + 3 𝑘﷯, 𝑏﷯ = 2 𝑖﷯ + 𝑗﷯ − 𝑘﷯, and 𝑐﷯ = 2 𝑖﷯ + 𝑗﷯ − 𝑘﷯, Given, 𝑎﷯ = 𝑖﷯ − 2 𝑗﷯ + 3 𝑘﷯ , 𝑏﷯ = 2 𝑖﷯ + 𝑗﷯ – 𝑘﷯ , 𝑐﷯ = 2 𝑖﷯ + 𝑗﷯ – 𝑘﷯ Volume of parallelepiped = 𝒂﷯ 𝒃﷯ 𝒄﷯﷯ = 1﷮−2﷮3﷮2﷮1﷮−1﷮2﷮1﷮−1﷯﷯ = 1 1×−1﷯−(1×−1)﷯ − (−2) 2×−1﷯−(2×−1)﷯ + 3 2×1﷯−(2×1)﷯ = 1 −1+1﷯+2 −2+2﷯+3 2−2﷯ = 1(0) + 2 (0) + 3(0) = 0