First slide

Vector triple product

Question

$The values of k for which the points A (1, 0, 3), B (- 1, 3, 4), C (1, 2, 1) and D (k .2, 5) are coplanar, are$

Moderate

A

1
B

2
C

0
D

-1

Solution

$Let a = (1, 0, 3), b = (- 1, 3, 4), c = (1, 2, 1) and d = (k, 2, 5) . Since A, B, C and D$
$are coplanar, we have [d b c] + [d c a] + [d a b] = [a b c]$
$\begin{array}{l} \Rightarrow |\begin{matrix} k \\ - 1 \\ 1 \end{matrix} \begin{matrix} 2 \\ 3 \\ 2 \end{matrix} \begin{matrix} 5 \\ 4 \\ 1 \end{matrix}| + |\begin{matrix} k \\ 1 \\ 1 \end{matrix} \begin{matrix} 2 \\ 2 \\ 0 \end{matrix} \begin{matrix} 5 \\ 1 \\ 3 \end{matrix}| + |\begin{matrix} k & 2 & 5 \\ 1 & 0 & 3 \\ - 1 & 3 & 4 \end{matrix}| = |\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix} \begin{matrix} 0 \\ 3 \\ 2 \end{matrix} \begin{matrix} 3 \\ 4 \\ 1 \end{matrix}| \\ \Rightarrow - 5 k - 15 + 6 k - 14 - 9 k + 1 = - 20 \\ \Rightarrow - 8 k - 28 = - 20 \Rightarrow k = - 1 \end{array}$

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