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UPSC NDA Math Model Paper 3

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Question : 74 of 120

Marks: +1, -0

What is the value of

\lambda

(\lambda \hat{i}+\hat{j}-\hat{k}) \times (3 \hat{i}-2 \hat{j}+4 \hat{k})

= (2 \hat{i}-11 \hat{j}-7 \hat{k}) ?

2
-2
1
7

Solution:

Consider,

A = (\lambda \hat{i}+\hat{j}-\hat{k}) = (\lambda, 1, -1)

and

B = (3 \hat{i}-2 \hat{j}+4 \hat{k}) = (3, -2, 4)

and

C = (2 \hat{i}-11 \hat{j}-7 \hat{k}) = (2, -11, -7)

Given

A \times B = C

We know that, Let

A = (u_1, u_2, u_3)

and

B = (v_1, v_2, v_3)

be any two vectors then its cross product is

A \times B: (u_1, u_2, u_3) \times (v_1, v_2, v_3)

= (u_2 v_3 - u_3 v_2, u_3 v_1 - u_1 v_3, u_1 v_2 - u_2 v_1)

Consider

A \times B = C

A \times B: (\lambda, 1, -1) \times (3, -2, 4)

= ((1)(4) - (-1)(-2), (-1)(3)

- (\lambda)(4), (\lambda)(-2) - (1)(3)) = (2, -11, -7)

\Rightarrow (2, -3 - 4\lambda, -2\lambda - 3)

= (2, -11, -7)

Vectors are equal. Their corresponding elements are equal.

\Rightarrow -2\lambda - 3 = -7

\Rightarrow 2\lambda + 3 = 7

\Rightarrow \lambda = 2

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