If $\hat{\mathbf{a}}, \mathbf{b}$ and $\hat{\boldsymbol{c}}$ are non-coplanar vectors and if $\mathbf{d}$ is…

If $\hat{\mathbf{a}}, \mathbf{b}$ and $\hat{\boldsymbol{c}}$ are non-coplanar vectors and if $\mathbf{d}$ is such that $\hat{\mathbf{d}}=\frac{1}{x}(\hat{\mathbf{a}}+\hat{\mathbf{b}}+\hat{\mathbf{c}})$ and $\hat{\mathbf{d}}=\frac{1}{y}(\hat{\mathbf{b}}+\hat{\mathbf{c}}+\hat{\mathbf{d}})$ where $x$ and $y$ are non-zero real numbers, then $\frac{1}{x y}(\hat{\mathbf{a}}+\hat{\mathbf{b}}+\hat{\mathbf{c}}+\hat{\mathbf{d}})$ equals to
  1. $3 c$
  2. $-a$
  3. $0$
  4. $2a$

Solution

Given, $\mathbf{d}=\frac{1}{x}(\mathbf{a}+\mathbf{b}+\mathbf{c})$ $ \begin{aligned} & \text { and } \quad \mathbf{d}=\frac{1}{y}(\mathbf{b}+\mathbf{c}+\mathbf{d}) \\ & \therefore \quad \mathrm{a}+\mathrm{b}+\mathrm{c}-\mathrm{xd}=0 \\ & \text { and } \quad \mathbf{b}+\mathbf{c}+\mathbf{d}-\mathrm{yd}=0 \\ & \Rightarrow \quad \mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}=0 \\ & \therefore \frac{1}{x y}(\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d})=\frac{1}{x y}(0)=0 \\ & \end{aligned} $

Asked in: AP EAMCET 2014

Practice more Vectors questions on Aicharya