Let a = i + j + k, b = i - j + k and c = i - j - k be three vectors. A vector v in the plane of a and b, whose projection on c is frac1√(3), is given by

Let a = i + j + k, b = i - j + k and c = i - j - k be three… — JEE Main 2011 Mathematics PYQ

AIEEE 2011

Let

\vec{a} = \hat{i} + \hat{j} + \hat{k}

\vec{b} = \hat{i} - \hat{j} + \hat{k}

and

\vec{c} = \hat{i} - \hat{j} - \hat{k}

be three vectors. A vector

\vec{v}

in the plane of

\vec{a}

and

\vec{b}

, whose projection on

\vec{c}

\frac{1}{\sqrt{3}}

, is given by

A.
$\hat{i} - 3\hat{j} + 3\hat{k}$
B.
$-3\hat{i} - 3\hat{j} - \hat{k}$
C.
$3\hat{i} - \hat{j} + 3\hat{k}$
D.
$\hat{i} + 3\hat{j} - 3\hat{k}$

Show answer and solution

Correct option: C

Solution

v lies in the plane of a and b, so v = λa + μb = (λ+μ)i + (λ-μ)j + (λ+μ)k.

Let p = λ+μ and q = λ-μ, so v = pi + qj + pk (note the i and k components are always equal).

Projection of v on c = (v·c)/|c| = (p - q - p)/√3 = (-q)/√3 = 1/√3, giving q = -1.

So v = pi - j + pk. Checking option 3: 3i - j + 3k has p=3, q=-1 ✓, and the i and k components are both 3 ✓.

[AIEEE 2011]

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