First slide

Introduction to vectors

Question

$The vectors \vec{a} = \hat{i} + \hat{j} + m \hat{k}, \vec{b} = \hat{i} + \hat{j} + (m + 1) \hat{k} and \vec{c} = \hat{i} - \hat{j} + m \hat{k} are coplanar if m is =$

Easy

A

1
B

4
C

3
D

None of these

Solution

$\vec{a}, \vec{b}, \vec{c} are coplanar if$
$\begin{array}{l} |\begin{matrix} 1 & 1 & m \\ 1 & 1 & m + 1 \\ 1 & - 1 & m \end{matrix}| = 0 \Rightarrow |\begin{matrix} 1 & 0 & m \\ 1 & 0 & m + 1 \\ 1 & - 2 & m \end{matrix}| = 0 \\ (c_{2} \to c_{2} - c_{1}) \\ \Rightarrow 2 = 0 \end{array}$
$So there is no value of m for which the vectors are coplanar.$

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