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UPSC NDA Math Model Paper 7

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Question : 47 of 120

Marks: +1, -0

If

\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2\hat{i}-4\hat{k},

\vec{c}=\hat{i}+\lambda\hat{j}+3\hat{k}

are coplanar, then the value of

\lambda

is:

5/2
3/5
7/3
5/3

Solution:

\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2\hat{i}-4\hat{k},

\vec{c}=\hat{i}+\lambda\hat{j}+3\hat{k}

are coplanar

As we know that, if three non-zero vectors

\vec{a}, \vec{b}

and

\vec{c}

are coplanar if and only if

[a\;\;b\;\;c]=0

We also know that,

\vec{a}\cdot(\vec{b}\times\vec{c})=\begin{vmatrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{vmatrix}

=\begin{bmatrix} a & b & c \end{bmatrix}

Here,

a_{1}=1, b_{1}=1, c_{1}=1, a_{2}=2, b_{2}=0,

c_{2}=-4, a_{3}=1, b_{3}=\lambda

and

c_{3}=3

\Rightarrow\begin{bmatrix} a & b & c \end{bmatrix}=\begin{vmatrix} 1 & 2 & 1 \\ 1 & 0 & \lambda \\ 1 & -4 & 3 \end{vmatrix}

=0

∴ 1(4λ) - 2(3 - λ) + 1(- 4) = 0

∴ 4λ - 6 + 2λ - 4 = 0

∴ λ = 5/3

∴ option (D) is the correct answer.

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