Example 27 - If i+j+k, 2i+5j, 3i+2j-3k and i-6j-k are position - Collinearity Of two vectors

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  1. Chapter 10 Class 12 Vector Algebra
  2. Serial order wise
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Example 27 If 𝑖﷯ + 𝑗﷯ + 𝑘﷯, 2 𝑖﷯ + 5 𝑗﷯, 3 𝑖﷯ + 2 𝑗﷯ – 3 𝑘﷯ and 𝑖﷯ – 6 𝑗﷯ – 𝑘﷯ are the position vectors of points A, B, C and D respectively, then find the angle between 𝐴𝐵﷯ and 𝐶𝐷﷯ . Deduce that 𝐴𝐵﷯ and 𝐶𝐷﷯ are collinear. Angle between 𝐴𝐵﷯ & 𝐶𝐷﷯ is given by cos θ = 𝑨𝑩﷯ . 𝑪𝑫﷯﷮ 𝑨𝑩﷯﷯ 𝑪𝑫﷯﷯﷯ Now, cos θ = 𝐴𝐵﷯ . 𝐶𝐷﷯﷮ 𝐴𝐵﷯﷯ 𝐶𝐷﷯﷯﷯ = 𝑖﷯ + 4 𝑗﷯ − 𝑘﷯﷯. −2 𝑖﷯ − 8 𝑗﷯ + 2 𝑘﷯﷯﷮3 ﷮2﷯ × 6 ﷮2﷯﷯ = 1 −2﷯ + 4 −8﷯ − 1(2)﷮3 ﷮2﷯ × 6 ﷮2﷯﷯ = −2 − 32 − 2﷮3× 6 × ﷮2﷯ × ﷮2﷯﷯ = −36﷮36﷯ = –1 Since cos θ = –1, θ = 180° So, θ = 180° = 180° × 𝝅﷮𝟏𝟖𝟎﷯ = π So, angle between 𝑨𝑩﷯ & 𝑪𝑫﷯ is π Also, Since angle between 𝐴𝐵﷯ & 𝐶𝐷﷯ is 180° , they are in opposite directions Since 𝐴𝐵﷯ & 𝐶𝐷﷯ are parallel to the same line 𝑚﷯, they are collinear.

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