For a non-zero real number $x$, if the points with position vectors $(x-u) \hat{\mathbf{i}}+x \hat{\mathbf{j}}+x \hat{\mathbf{k}}, x \hat{\mathbf{i}}+(x-v) \hat{\mathbf{j}}+x \hat{\mathbf{k}}$, $x \hat{\mathbf{i}}+x \hat{\mathbf{j}}+(x-w) \hat{\mathbf{k}}$ and \((x-1) \hat{\mathbf{i}}+(x-1)…

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For a non-zero real number $x$, if the points with position vectors $(x-u) \hat{\mathbf{i}}+x \hat{\mathbf{j}}+x \hat{\mathbf{k}}, x \hat{\mathbf{i}}+(x-v) \hat{\mathbf{j}}+x \hat{\mathbf{k}}$, $x \hat{\mathbf{i}}+x \hat{\mathbf{j}}+(x-w) \hat{\mathbf{k}}$ and $(x-1) \hat{\mathbf{i}}+(x-1) \hat{\mathbf{j}}+(x-1) \hat{\mathbf{k}}$ are coplanar, then

Aicharya · Accepted Answer

Let $\begin{aligned} & \mathbf{O A}=(x-u) \hat{\mathbf{i}}+x \hat{\mathbf{j}}+x \hat{\mathbf{k}} \ & \mathbf{O B}=x \hat{\mathbf{i}}+(x-v) \hat{\mathbf{j}}+x \hat{\mathbf{k}} \ & \mathbf{O C}=x \hat{\mathbf{i}}+x \hat{\mathbf{j}}+(x-w) \hat{\mathbf{k}} \ & \mathbf{O D}=(x-1) \hat{\mathbf{i}}+(x-1) \hat{\mathbf{j}}+(x-1) \hat{\mathbf{k}} \end{aligned}$ and $\quad \mathbf{O D}=(x-1) \hat{\mathbf{i}}+(x-1) \hat{\mathbf{j}}+(x-1) \hat{\mathbf{k}}$ Here, $\quad \mathbf{D A}=(\mathbf{l}-u) \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ $\mathbf{D B}=\hat{\mathbf{i}}+(\mathbf{l}-v) \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{D C}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+(\mathbf{l}-w) \hat{\mathbf{k}}$ Since, points are collinear [DA DB DC] $=0$ $\Rightarrow \quad\left[\begin{array}{ccc} 1-u & 1 & 1 \ 1 & 1-v & 1 \ 1 & 1 & 1-w \end{array}\right]=0$ On solving this, we get $\frac{1}{u}+\frac{1}{v}+\frac{1}{w}=1$

For a non-zero real number \(x\), if the points with position vectors \((x-u) \hat{\mathbf{i}}+x…

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