Question

If vectors ${\sec }^{2}A\hat{i}+\hat{j}+\hat{k},\hat{i}+{\sec }^{2}B\hat{j}+\hat{k}$, and $\hat{i}+\hat{j}+{\sec }^2\hat{k}$ are coplanar, then ${\cot }^{2}A+{\cot }^{2}B+{\cot }^{2}C$ is

• A. equal to $1$
• B. equal to $2$
• C. equal to $0$
• D. none of these

Answer 1

The correct option is A equal to $1$

Since vector ${\sec }^2\hat{i}+\hat{j}+\hat{k}/\hat{i}+{\sec }^2\hat{j}+\hat{k};\hat{i}+\hat{j}+{\sec }^2\hat{k}$ are coplaner

$\left|\begin{array}{ccc}{\sec }^{2}A& 1& 1\\ 1& {\sec }^{2}B& 1\\ 1& 1& {\sec }^{2}C\end{array}\right|=0$

$\therefore {\sec }^{2}A({\sec }^{2}B{\sec }^{2}C-1)-1({\sec }^{2}C-1)+1(1-{\sec }^{2}B)$

$\therefore {\sec }^{2}A{\sec }^{2}B{\sec }^{2}C-{\sec }^{2}A-{\sec }^{2}C+1+1-{\sec }^{2}B=-0$

$\therefore {\sec }^{2}A{\sec }^{2}B{\sec }^{2}C-{\sec }^{2}A-{\sec }^{2}B.{\sec }^{2}C=-2$

$\therefore {\cot }^{2}A+{\cot }^{2}B+{\cot }^{2}C=1$