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Example 25 - Find area of a parallelogram whose a = 3i + j + 4k - Vector product - Area

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Example 25 Find the area of a parallelogram whose adjacent sides are given by the vectors a = 3 𝑖﷯ + 𝑗﷯ + 4 𝑘﷯ and b = 𝑖﷯ − 𝑗﷯ + k﷯ a = 3 𝑖﷯ + 1 𝑗﷯ + 4 𝑘﷯ b = 1 𝑖﷯ − 1 𝑗﷯ + 1 k﷯ Area of parallelogram ABCD = 𝑎﷯× 𝑏﷯﷯ 𝒂﷯× 𝒃﷯ = 𝑖﷯﷮ 𝑗﷯﷮ 𝑘﷯﷮3﷮1﷮4﷮1﷮−1﷮1﷯﷯ = 𝑖﷯ (1 × 1 – (−1) × 4) − 𝑗﷯ (3 × 1 – 1 × 4) + 𝑘﷯ (3 × −1 − 1 × 1) = 𝑖﷯(1 − (-4)) − j﷯ (3 − 4) + 𝑘﷯ (−3 −1) = 𝑖﷯(1 + 4) − j﷯ (−1) + 𝑘﷯ (−4) = 5 𝒊﷯ + 𝒋﷯ − 4 𝒌﷯ Magnitude of 𝑎﷯× 𝑏﷯ = ﷮52+ 1﷮2﷯+ −4﷯2﷯ 𝑎﷯× 𝑏﷯﷯ = ﷮25+1+16﷯ = ﷮42﷯ Area of parallelogram ABCD = 𝑎﷯× 𝑏﷯﷯ = ﷮42﷯ Therefore, the required area is ﷮𝟒𝟐﷯ .

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