Let $\mathbf{A}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$. If $\mathbf{B}$ is a vector in $X Y$ plane such that…

Let $\mathbf{A}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$. If $\mathbf{B}$ is a vector in $X Y$ plane such that $(\mathbf{A}+\mathbf{B}) \cdot \mathbf{B}=15$ and $\mathbf{A} \cdot \mathbf{B}=6$, then $|\mathbf{B}|$ is
  1. $6$
  2. $9$
  3. $15$
  4. $3$

Solution

$\mathbf{A}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$ Let $\mathbf{B}=x \hat{\mathbf{i}}+y \hat{\mathbf{j}}$ $\because(\mathbf{A}+\mathbf{B}) \cdot \mathbf{B}=15$ and $\mathbf{A} \cdot \mathbf{B}=6$ $\Rightarrow \mathbf{A} \cdot \mathbf{B}+\mathbf{B} \cdot \mathbf{B}=15$ $\begin{array}{cc}\Rightarrow & 6+|\mathbf{B}|^2=15 \\ \Rightarrow & |\mathbf{B}|^2=15-6=9 \\ \Rightarrow & |\mathbf{B}|=3\end{array}$

Asked in: AP EAMCET 2021 (23 Aug Shift 2)

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