Given, \(\triangle A B C\) such that \(A\) is \(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, B\) is…
Given, \(\triangle A B C\) such that \(A\) is \(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, B\) is \(\hat{\mathbf{i}}-3 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}\) and \(C\) is \(3 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}\), then \(\triangle A B C\) is
An equilateral triangle
A right-angled triangle
An isosceles triangle
A scalene triangle
Solution
Vertices of \(\triangle A B C\) are given as
\(A\) is \(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\)
\(B\) is \(\hat{\mathbf{i}}-3 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}\)
and \(C\) is \(3 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}\)
\(\therefore \quad \mathbf{A B}=-\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-6 \hat{\mathbf{k}}, \mathbf{B C}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\)
and \(\quad \mathbf{C A}=\hat{\mathbf{i}}-3 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}\)
\(\begin{gathered}
\because \quad|\mathbf{A B}|=\sqrt{1+4+36}=\sqrt{41} \\
|\mathbf{B C}|=\sqrt{4+1+1}=\sqrt{6}
\end{gathered}\)
and \(|\mathbf{C A}|=\sqrt{1+9+25}=\sqrt{35}\)
\(\because \quad|\mathbf{A B}|^2=|\mathbf{B C}|^2+|\mathbf{C A}|^2\)
\(\therefore \triangle A B C\) is an right-angled triangle.
Hence, option (b) is correct.