NIMCET 2025 — Mathematics PYQ

To find the required vector, we need to determine the vector parallel to the linear combination $2\vec{a}-\vec{b}+3\vec{c}$ and then scale it to the specified magnitude of $\sqrt{22}$.

Step 1: Compute the Resultant Vector $\vec{r}$

Let $\vec{r}=2\vec{a}-\vec{b}+3\vec{c}$.

Substitute the given component values of the vectors:

$$\vec{r}=2\left(\hat{i}+\hat{j}+\hat{k}\right)-\left(2\hat{i}-\hat{j}+3\hat{k}\right)+3\left(\hat{i}-2\hat{j}+\hat{k}\right)$$

Expand the scalar multiplications:

$$\vec{r}=\left(2\hat{i}+2\hat{j}+2\hat{k}\right)-\left(2\hat{i}-\hat{j}+3\hat{k}\right)+\left(3\hat{i}-6\hat{j}+3\hat{k}\right)$$

Group the corresponding components together along $\hat{i}$, $\hat{j}$, and $\hat{k}$:

$$\vec{r}=\left(2-2+3\right)\hat{i}+\left(2-(-1)+3(-2)\right)\hat{j}+\left(2-3+3\right)\hat{k}$$

Simplify each bracket:

$$\vec{r}=3\hat{i}+\left(2+1-6\right)\hat{j}+2\hat{k}$$

$$\vec{r}=3\hat{i}-3\hat{j}+2\hat{k}$$

Step 2: Calculate the Magnitude of $\vec{r}$

The magnitude of vector $\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$ is given by $|\vec{r}|=\sqrt{x^2+y^2+z^2}$:

$$|\vec{r}|=\sqrt{3^2+(-3{)}^2+2^2}$$

$$|\vec{r}|=\sqrt{9+9+4}$$

$$|\vec{r}|=\sqrt{22}$$

Step 3: Find the Required Parallel Vector

A vector of magnitude $M$ that is parallel to any given vector $\vec{r}$ is calculated using the formula:

$$\vec{V}=\pm M\cdot \hat{r}=\pm M\cdot \frac{\vec{r}}{|\vec{r}|}$$

Substitute $M=\sqrt{22}$, $\vec{r}=3\hat{i}-3\hat{j}+2\hat{k}$, and $|\vec{r}|=\sqrt{22}$:

$$\vec{V}=\pm \sqrt{22}\cdot \frac{3\hat{i}-3\hat{j}+2\hat{k}}{\sqrt{22}}$$

Canceling $\sqrt{22}$ from the numerator and denominator gives:

$$\vec{V}=\pm \left(3\hat{i}-3\hat{j}+2\hat{k}\right)$$

Correct Answer

Correct Option: (A)