JEE Mathematics 2026 Question with Solution

Given: $\vec{a}=2\hat{i}-\hat{j}-\hat{k}$, $\vec{b}=\hat{i}+3\hat{j}-\hat{k}$, $\vec{c}=2\hat{i}+\hat{j}+3\hat{k}$. The length of projection of $\vec{v}$ on $\vec{c}$ is $\dfrac{1}{\sqrt{14}}$, and $\vec{v}$ lies in the plane of $\vec{a}$ and $\vec{b}$.

Find: $|\vec{v}|$.

Use the projection formula:

$$\frac{|\vec{v}\cdot\vec{c}|}{|\vec{c}|}=\frac{1}{\sqrt{14}}$$

Now compute the magnitude of $\vec{c}$:

$$|\vec{c}|=\sqrt{2^2+1^2+3^2}=\sqrt{4+1+9}=\sqrt{14}$$

Therefore,

$$\frac{|\vec{v}\cdot\vec{c}|}{\sqrt{14}}=\frac{1}{\sqrt{14}}$$

So,

$$|\vec{v}\cdot\vec{c}|=1$$

Since $\vec{v}$ is in the plane of $\vec{a}$ and $\vec{b}$, write

$$\vec{v}=m\vec{a}+n\vec{b}$$

Hence,

$$\vec{v}=m(2\hat{i}-\hat{j}-\hat{k})+n(\hat{i}+3\hat{j}-\hat{k})$$ $$\vec{v}=(2m+n)\hat{i}+(-m+3n)\hat{j}+(-m-n)\hat{k}$$

Take dot product with $\vec{c}=2\hat{i}+\hat{j}+3\hat{k}$:

$$\vec{v}\cdot\vec{c}=2(2m+n)+(-m+3n)+3(-m-n)$$ $$\vec{v}\cdot\vec{c}=4m+2n-m+3n-3m-3n$$ $$\vec{v}\cdot\vec{c}=5n$$

This algebra from direct substitution gives $\vec{v}\cdot\vec{c}=5n$, whereas the extracted solution states $\vec{v}\cdot\vec{c}=7(m+n)$ and concludes the correct option is C. Based on the provided the solution, the accepted answer is C.

Therefore, the correct option is C, so $|\vec{v}|=7$ according to the source solution.

Given: The source solution claims $\vec{v}\cdot\vec{c}=7(m+n)$ after writing $\vec{v}=m\vec{a}+n\vec{b}$.

Find: Whether that intermediate step matches the listed vectors.

Using the given vectors,

$$\vec{a}\cdot\vec{c}=(2)(2)+(-1)(1)+(-1)(3)=4-1-3=0$$ $$\vec{b}\cdot\vec{c}=(1)(2)+(3)(1)+(-1)(3)=2+3-3=2$$

So,

$$\vec{v}\cdot\vec{c}=(m\vec{a}+n\vec{b})\cdot\vec{c}=m(\vec{a}\cdot\vec{c})+n(\vec{b}\cdot\vec{c})=0+2n=2n$$

This does not match the extracted line $\vec{v}\cdot\vec{c}=7(m+n)$. Therefore, the working shown on the page appears internally inconsistent with the printed vectors, even though the page explicitly marks C as the correct option.

Because the instruction is to treat the solution, the final recorded answer remains C.