NIMCET 2021 — Mathematics PYQ

• Angle $\theta$ between $\vec{a}$ and $\vec{b}$ is $120^{\circ }$.

• $|\vec{b}|=2|\vec{a}|$. Let $|\vec{a}|=k$, then $|\vec{b}|=2k$.

• The dot product of $\vec{a}$ and $\vec{b}$ is:

  $$\vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|\cos (120^{\circ })$$
  $$\vec{a}\cdot \vec{b}=(k)(2k)\left(-\frac{1}{2}\right)=-k^2$$

Step 2: Use the condition for perpendicular vectors.

Two vectors are at right angles (perpendicular) if their dot product is zero.

Expanding the dot product:

Step 3: Substitute the values in terms of $k$.

We know:

• $|\vec{a}{|}^2=k^2$

• $|\vec{b}{|}^2=(2k{)}^2=4k^2$

• $\vec{a}\cdot \vec{b}=-k^2$

Substituting these into the equation:

Step 4: Solve for $x$.

Since $\vec{a}$ and $\vec{b}$ are non-zero vectors, $k\ne 0$. Dividing the equation by $k^2$:

Final Answer:

The value of $x$ is $\frac{2}{5}$ (or $0.4$).