Ex 10.2, 19 - If a,  bare two collinear vectors, then which - Collinearity Of two vectors

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  1. Chapter 10 Class 12 Vector Algebra
  2. Serial order wise
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Ex 10.2, 19 If 𝑎﷯ and 𝑏﷯ are two collinear vectors, then which of the following are incorrect: (A) 𝑏﷯ = λ 𝑎﷯, for some scalar λ (B) 𝑎﷯ = ± 𝑏﷯ (C) the respective components of 𝑎﷯ and 𝑏﷯ are not proportional (D) both the vectors 𝑎﷯ and 𝑏﷯ have same direction, but different magnitudes. Given, a﷯ and b﷯ are collinear vectors Checking (A) 𝑏﷯ = λ 𝑎﷯ Let 𝑎﷯ = 1 𝑖﷯ + 1 𝑗﷯ + 1 𝑘﷯ & 𝑏﷯ = 3 𝑖﷯ + 3 𝑗﷯ + 3 𝑘﷯ Where 𝑎﷯ & 𝑏﷯ are collinear (direction ratio are proportional) b﷯ = 3 i﷯ + 3 j﷯ + 3 k﷯ = 3 (1 i﷯ + 1 j﷯ + 1 k﷯) = 3 a﷯ So, b﷯ = 𝜆 a﷯ , where 𝜆 = 3 is a scalar. Hence, (A) is correct Checking (B) 𝑎﷯ = ± 𝑏﷯ Since 𝑎﷯ & 𝑏﷯ are collinear Let 𝑎﷯ = 1 i﷯ + 1 j﷯ + 1 k﷯ So, 𝑎﷯ = ± b﷯ Hence, (B) is correct Checking (C) the respective components of a﷯ and b﷯ are not proportional Since 𝑎﷯ & 𝑏﷯ are collinear Let 𝑎﷯ = 1 𝑖﷯ + 2 𝑗﷯ + 3 𝑘﷯ & 𝑏﷯ = 4 𝑖﷯ + 8 𝑗﷯ + 12 𝑘﷯ Ratio of respective components = 1﷮4﷯ , 2﷮8﷯ , 3﷮12﷯ = 1﷮4﷯ , 1﷮4﷯ , 1﷮4﷯ So, respective components of a﷯ & b﷯ are proportional. Hence, (C) is correct. Checking (D) both the vectors 𝑎﷯ and 𝑏﷯ have same direction, but different magnitudes. Since 𝑎﷯ & 𝑏﷯ are collinear Let 𝑎﷯ = 1 𝑖﷯ + 1 𝑗﷯ + 1 𝑘﷯ & 𝑏﷯ = –2 𝑖﷯ – 2 𝑗﷯ – 2 𝑘﷯ Magnitude of 𝑎﷯ = ﷮12+12+12﷯ 𝑎﷯﷯ = ﷮1+1+1﷯ = ﷮3﷯ Magnitude of b﷯ = ﷮ −2﷯2+(−2)2+(−2)2﷯ 𝑏﷯﷯ = ﷮4+4+4﷯ = ﷮12﷯ = ﷮4×3﷯ = 2 ﷮3﷯ Thus, 𝑏﷯﷯ = 2 𝑎﷯﷯ i.e.. they have different magnitudes So, a﷯ & b﷯ have different magnitudes Also, b﷯ = −2 (1 i﷯ + 1 j﷯ + 1 k﷯) = –2 𝑎﷯ ∴ 𝑏﷯ and 𝑎﷯ have opposite directions Hence, (D) is incorrect. So, D is the correct option

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