First slide

Planes in 3D

Question

If the plan $\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = q_{1}, \vec{r} \cdot (\hat{i} + 2 a \hat{j} + \hat{k}) = q_{2} and \vec{r} \cdot (a \hat{i} + a^{2} \hat{j} + \hat{k}) = q_{3}$ intersect in a line, then the value of a is

Moderate

A

1
B

½
C

2
D

0

Solution

$\vec{r} \cdot {\vec{n}}_{1} = {\vec{q}}_{1} and \vec{r} \cdot {\vec{n}}_{2} = {\vec{q}}_{2}, \vec{r} \cdot {\vec{n}}_{3} = {\vec{q}}_{3}$ intersect in a line if $[\begin{array}{lll} {\vec{n}}_{1} {\vec{n}}_{2} {\vec{n}}_{3} \end{array}] = 0$ ,So
$\begin{array}{l} |\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 2 a & 1 \\ a & a^{2} & 1 \end{array}| = 0 \\ or 2 a - a^{2} - 1 + a + a^{2} - 2 a^{2} = 0 \\ or 2 a^{2} - 3 a + 1 = 0 \\ or a = 1 / 2, 1 \end{array}$

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