Let a = 2i + j - 2k, b = i + j and c be a vector such that |c - a| = 3, |(a × b) × c| = 3 and the angle between c and a × b is 30°. Then, a · c is equal to

Let a = 2i + j - 2k, b = i + j and c be a vector such that… — JEE Main 2017 Mathematics PYQ

JEE Main 2017

Let

\mathbf{a} = 2\hat{i} + \hat{j} - 2\hat{k}

\mathbf{b} = \hat{i} + \hat{j}

and

\mathbf{c}

be a vector such that

|\mathbf{c} - \mathbf{a}| = 3

|(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}| = 3

and the angle between

\mathbf{c}

and

\mathbf{a} \times \mathbf{b}

30°

. Then,

\mathbf{a} \cdot \mathbf{c}

is equal to

A.
$\frac{25}{8}$
B.
$2$
C.
$5$
D.
$\frac{1}{8}$

Show answer and solution

Correct option: B

Solution

Compute a×b: a = (2,1,-2), b = (1,1,0).
a×b = |i j k; 2 1 -2; 1 1 0| = i(0+2) - j(0+2) + k(2-1) = (2,-2,1).
|a×b| = 3.

|(a×b)×c| = |a×b||c|sin30° = 3|c|·(1/2) = (3/2)|c| = 3, so |c| = 2.

Now |c - a|² = 9: |c|² - 2(a·c) + |a|² = 9.
|a|² = 4+1+4 = 9, |c|² = 4.
4 - 2(a·c) + 9 = 9 → 2(a·c) = 4 → a·c = 2.

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