Ex 10.4, 8 - Chapter 10 Class 12 Vector Algebra

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Ex 10.4, 8 If either 𝑎﷯ = 0﷯ or 𝑏﷯ = 0﷯, then 𝑎﷯ × 𝑏﷯ = 0﷯ . Is the converse true? Justify your answer with an example. Converse : If 𝑎﷯ × 𝑏﷯ = 0﷯, then either 𝑎﷯ = 0﷯ or 𝑏﷯ = 0﷯ 𝑎﷯ × 𝑏﷯ = 𝑎﷯﷯ 𝑏﷯﷯ sin θ 𝑛﷯ where, θ = angle between 𝑎﷯ and 𝑏﷯ 𝑛﷯ = unit vector perpendicular to 𝑎﷯ 𝜀 𝑏﷯ Let 𝑎﷯ = 1 𝑖﷯ + 1 𝑗﷯ + 1 𝑘﷯ & 𝑏﷯ = 2 𝑖﷯ + 2 𝑗﷯ + 2 𝑘﷯ 𝑎﷯ × 𝑏﷯ = 𝑖﷯﷮ 𝑗﷯﷮ 𝑘﷯﷮1﷮1﷮1﷮2﷮2﷮2﷯﷯ = 𝑖﷯ (1 × 2 − 2 × 1) − 𝑗﷯ (1 × 2 −2 × 1) + 𝑘﷯ (1 × 2 − 2 × 1) + 𝑘﷯(1 × 2 − 2 × 1) = 𝑖﷯ (2 − 2) − 𝑗﷯ (2 −2) + 𝑘﷯ (2 − 2) = 0 𝑖﷯ − 0 𝑗﷯ + 0 𝑘﷯ = 0﷯ Here, 𝑎﷯ ≠ 0﷯ & 𝑏﷯≠ 0﷯ But 𝑎﷯ × 𝑏﷯ = 0﷯ Therefore, converse is not true.