JEE 2023 — Mathematics PYQ

To find the value of $\vec{b}\cdot \vec{c}$, we substitute the expression for $\vec{c}$ given in the problem:

1. Substitute $\vec{c}$ into the dot product:

2. Distribute the dot product across the terms:

3. Evaluate the terms individually:

• The first term: $\vec{b}\cdot (\vec{a}\times \vec{b})$. By definition, the cross product $(\vec{a}\times \vec{b})$ produces a vector that is perpendicular to both $\vec{a}$ and $\vec{b}$. Since it is perpendicular to $\vec{b}$, their dot product must be zero.

  $$2(\vec{b}\cdot (\vec{a}\times \vec{b}))=2(0)=0$$

• The second term: $\vec{b}\cdot \vec{b}$ is equivalent to the square of the magnitude, $|\vec{b}{|}^2$. Given that $|\vec{b}|=4$:

  $$3(\vec{b}\cdot \vec{b})=3(4^2)=3(16)=48$$

4. Combine the results:

Final Answer:

The correct option is (3) -48.