The vectors a-mk- b - -m-1k- and c-mk- are coplanar- if m is equal to-

Explanation for correct option Given vector are a=î+ĵ+mk̂, b = î+ĵ+(m+1)k̂ and c=î ĵ+mk̂We know that isa, b and c are co planer then the determinant of coeffic ...

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Byju's Answer

Standard XII

Mathematics

Test for Coplanarity

The vectors a...

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 equal to?

$1$

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$4$

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$3$

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no value of $m$ for which vectors are coplanar

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Solution

The correct option is D
no value of $m$ for which vectors are coplanar

Explanation for correct option
Given vector are
$\vec{a} = \hat{i} + \hat{j} + m \hat{k}, \vec{b} = \hat{i} + \hat{j} + (m + 1) \hat{k}$ and $\vec{c} = \hat{i} - \hat{j} + m \hat{k}$
We know that is $\vec{a}, \vec{b}$ and $\vec{c}$ are co-planer then the determinant of coefficient matrix is $0$
$∴ |\begin{matrix} 1 & 1 & m \\ 1 & 1 & m - 1 \\ 1 & - 1 & m \end{matrix}| = 0 \Rightarrow 1 (m + m - 1) + 1 + m (- 1 - 1) = 0 \Rightarrow 2 m - 1 + 1 - 2 m = 0$
Hence, option (D) is correct

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The vectors a→=i^+j^+mk^,b→=i^+j^+(m+1)k^ and c→=i^-j^+mk^ are coplanar, if m is equal to?

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 equal to?