NIMCET 2015 — Mathematics PYQ

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Question

NIMCET 2015 — Mathematics PYQ

NIMCET | Mathematics | 2015

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 $\frac{\vec{c}}{|\vec{c}|}$ is $\frac{1}{\sqrt{3}}$ , is:

Choose the correct answer:

Explanation

1. Express vector $\vec{v}$ in terms of $\vec{a}$ and $\vec{b}$

Since $\vec{v}$ lies in the plane of $\vec{a}$ and $\vec{b}$ , it can be written as a linear combination:

\vec{v} = \lambda \vec{a} + \mu \vec{b}

\vec{v} = \lambda(\hat{i} + \hat{j} + \hat{k}) + \mu(\hat{i} - \hat{j} + \hat{k})

\vec{v} = (\lambda + \mu)\hat{i} + (\lambda - \mu)\hat{j} + (\lambda + \mu)\hat{k}

2. Use the Projection Condition

The projection of $\vec{v}$ on $\vec{c}$ is given as $\frac{1}{\sqrt{3}}$ . The formula for the scalar projection of $\vec{v}$ on $\vec{c}$ is $\frac{\vec{v} \cdot \vec{c}}{|\vec{c}|}$ .

First, find $|\vec{c}|$ :

|\vec{c}| = \sqrt{1^2 + (-1)^2 + (-1)^2} = \sqrt{3}

According to the question:

\frac{\vec{v} \cdot \vec{c}}{\sqrt{3}} = \frac{1}{\sqrt{3}} \implies \vec{v} \cdot \vec{c} = 1

3. Calculate the Dot Product

Substitute $\vec{v}$ and $\vec{c} = \hat{i} - \hat{j} - \hat{k}$ :

((\lambda + \mu)\hat{i} + (\lambda - \mu)\hat{j} + (\lambda + \mu)\hat{k}) \cdot (\hat{i} - \hat{j} - \hat{k}) = 1

(\lambda + \mu)(1) + (\lambda - \mu)(-1) + (\lambda + \mu)(-1) = 1

\lambda + \mu - \lambda + \mu - \lambda - \mu = 1

-\lambda + \mu = 1 \implies \mu = \lambda + 1

4. Formulate $\vec{v}$ with one variable

Substitute $\mu = \lambda + 1$ back into the expression for $\vec{v}$ :

\vec{v} = (\lambda + \lambda + 1)\hat{i} + (\lambda - (\lambda + 1))\hat{j} + (\lambda + \lambda + 1)\hat{k}

\vec{v} = (2\lambda + 1)\hat{i} - \hat{j} + (2\lambda + 1)\hat{k}

5. Match with Options

We need to find a value of $\lambda$ that results in one of the given options.

If $\lambda = 1$ : $\vec{v} = (2(1) + 1)\hat{i} - \hat{j} + (2(1) + 1)\hat{k} = 3\hat{i} - \hat{j} + 3\hat{k}$
This matches option (a).

Correct Option: (a)