Let a = a_1i + a_2j + a_3k, b = b_1i + b_2j + b_3k and c = c_1i + c_2j + c_3k be three non-zero vectors such that c is a unit vector perpendicular to both the vectors a and b. If the angle between a and b is (π)/6, then beginvmatrix a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 endvmatrix^2 is equal to

1/4(a_1^2 + a_2^2 + a_3^2)(b_1^2 + b_2^2 + b_3^2)

3/4(a_1^2 + a_2^2 + a_3^2)(b_1^2 + b_2^2 + b_3^2)(c_1^2 + c_2^2 + c_3^2)

Let a = a_1i + a_2j + a_3k, b = b_1i + b_2j + b_3k and c =… — JEE Main 1986 Mathematics PYQ

1986

Let

\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}

\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}

and

\vec{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k}

be three non-zero vectors such that

\vec{c}

is a unit vector perpendicular to both the vectors

\vec{a}

and

\vec{b}

. If the angle between

\vec{a}

and

\vec{b}

\dfrac{\pi}{6}

, then

\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}^2

is equal to

A.
0
B.
1
C.
$\dfrac{1}{4}(a_1^2 + a_2^2 + a_3^2)(b_1^2 + b_2^2 + b_3^2)$
D.
$\dfrac{3}{4}(a_1^2 + a_2^2 + a_3^2)(b_1^2 + b_2^2 + b_3^2)(c_1^2 + c_2^2 + c_3^2)$

Show answer

Correct option: C

[1986]

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