NIMCET 2015 — Mathematics PYQ

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

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 $|\vec{c}|$:

According to the question:

3. Calculate the Dot Product

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

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

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

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)