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JEE Main · 2005 · 08 May 2005 · Offline · Vector Algebra

For any vector

a

, the value of

(a \times \hat{i})^{2} + (a \times \hat{j})^{2} + (a \times \hat{k})^{2}

is equal to

(A) $3 a^{2}$
(B) $a^{2}$
(C) $2 a^{2}$ ✓ Correct
(D) $4 a^{2}$

Answer: (C)

Solution

Write

a = x \hat{i} + y \hat{j} + z \hat{k}

. Then

a \times \hat{i} = z \hat{j} - y \hat{k}

,

a \times \hat{j} = x \hat{k} - z \hat{i}

and

a \times \hat{k} = y \hat{i} - x \hat{j}

, whose squared magnitudes are

y^{2} + z^{2}

,

z^{2} + x^{2}

and

x^{2} + y^{2}

. Adding these gives

2 (x^{2} + y^{2} + z^{2}) = 2 a^{2}

.

JEE Main · 2005 · 08 May 2005 · Offline · Single correct option

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