Let $$\overrightarrow a = \widehat i + 2\widehat j + \lambda \widehat k,\overrightarrow b = 3\widehat i - 5\widehat j - \lambda \widehat k,\overrightarrow a \,.\,\overrightarrow c = 7,2\overrightarrow b \,.\,\overrightarrow c + 43 = 0,\overrightarrow a \times \overrightarrow c = \overrightarrow b \times \overrightarrow c $$. Then $\left| {\overrightarrow a \,.\,\overrightarrow b } \right|$ is equal to :

$$ \begin{aligned} & \vec{a}=\hat{i}+2 \hat{j}+\lambda \hat{k}, \vec{b}=3 \hat{i}-5 \hat{j}-\lambda \hat{k}, \vec{a} \cdot \vec{c}=7 \\\\ & \vec{a} \times \vec{c}-\vec{b} \times \vec{c}=\overrightarrow{0} \\\\ & (\vec{a}-\vec{b}) \times \vec{c}=0 \Rightarrow(\vec{a}-\vec{b}) \text { is paralleled to } \vec{c} \\\\ & \vec{a}-\vec{b}=\mu \vec{c}, \text { where } \mu \text { is a scalar } \\\\ & -2 \hat{i}+7 \hat{\mathrm{j}}+2 \lambda \hat{k}=\mu \cdot \overrightarrow{\mathrm{c}} \end{aligned} $$<br/><br/> Now $\vec{a} \cdot \overrightarrow{\mathbf{c}}=7$ gives $2 \lambda^2+12=7 \mu$<br/><br/> And $\vec{b} \cdot \vec{c}=-\frac{43}{2}$ gives $4 \lambda^2+82=43 \mu$<br/><br/> $\mu=2$ and $\lambda^2=1$<br/><br/> $|\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}|=8$