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JEE Main · 2011 · 10 Apr 2011 · Offline · Vector Algebra

If

a = \frac{1}{10} (3 \hat{i} + \hat{k})

and

b = \frac{1}{7} (2 \hat{i} + 3 \hat{j} - 6 \hat{k})

, then the value of

(2 a - b) \cdot [(a \times b) \times (a + 2 b)]

is

(A)-3
(B)5
(C)3
(D)-5✓ Correct

Answer: (D)

Solution

Both

a

and

b

are unit vectors, and their dot product is

a \cdot b = \frac{1}{7 10} (3 \cdot 2 + 0 \cdot 3 + 1 \cdot (- 6)) = 0

, so they are perpendicular. Expand the triple product with

(u \times v) \times w = (u \cdot w) v - (v \cdot w) u

:

(a \times b) \times (a + 2 b) = (a \cdot (a + 2 b)) b - (b \cdot (a + 2 b)) a = b - 2 a

. Dotting with

2 a - b

,

(2 a - b) \cdot (b - 2 a) = 4 (a \cdot b) - 4∣ a ∣^{2} - ∣ b ∣^{2} = 0 - 4 - 1 = - 5

.

JEE Main · 2011 · 10 Apr 2011 · Offline · Single correct option

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