Show that the points whose position vectors are -2i+3j, i+2j+3k and 7i-k are collinear

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Solution

Suppose A, B and C are the points which are represented by;
$\vec{A} = - 2 i + 3 j; \vec{B} = i + 2 j + 3 k$ and $\vec{C} = 7 i - k$
Coordinates of these points are A(-2, 3, 0); B(1, 2, 3) and C(7, 0, -1).
So, $\vec{A B} = (1 + 2, 2 - 3, 3 - 0) = (3, - 1, 3) \vec{A C} = (7 + 2, 0 - 3, - 1 - 0) = (9, - 3, - 1)$
$|\begin{matrix} 3 & - 1 & 3 \\ 9 & - 3 & - 1 \end{matrix}| = 3 (1 + 9) + 1 (- 3 - 27) + 3 (- 9 + 9) = 0$
Therefore, points A, B and C are collinear.

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