Question
Let A = 3i + 2j, B = pi + 9j, and C = i + j. If points
A, B, and C are collinear, then the value of p is:Solution
We are given the position vectors of three points: A = 3i + 2j B = pi + 9j C = i + j To determine the value of p such that points A, B, and C are collinear, we use the condition that vectors AB and AC must be parallel, i.e., their cross product must be zero: AB Γ AC = 0 AB = B β A = (p β 3)i + (9 β 2)j = (p β 3)i + 7j AC = C β A = (1 β 3)i + (1 β 2)j = β2i β j Take the cross product in 2D (which is a scalar): In 2D, the cross product of vectors u = ai + bj and v = ci + dj is: u Γ v = ad β bc So: AB Γ AC = (p β 3)(β1) β (7)(β2) = β(p β 3) + 14 = βp + 3 + 14 = βp + 17 Set this equal to zero: βp + 17 = 0 β p = 17
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