Question

If $P,Q$ and $R$ are angles of $△PQR$, then the value of $\left|\begin{array}{ccc}-1& \cos R& \cos Q\\ \cos R& -1& \cos P\\ \cos Q& \cos P& -1\end{array}\right|$ is equal to

• A. $-1$
• B. $0$
• C. $\frac{1}{2}$
• D. $1$

Answer 1

The correct option is B $0$

Let $△=\left|\begin{array}{ccc}-1& \cos R& \cos Q\\ \cos R& -1& \cos P\\ \cos Q& \cos P& -1\end{array}\right|$

Multiplying $C_1$ by $p$ and then doing
$C_1\to C_1+q{C}_2+r{C}_3$, we get

$△=\frac{1}{p}\left|\begin{array}{ccc}-p& \cos R& \cos Q\\ p\cos R& -1& \cos P\\ p\cos Q& \cos P& -1\end{array}\right|$

$=\frac{1}{p}\left|\begin{array}{ccc}-p+q\cos R+r\cos Q& \cos R& \cos Q\\ p\cos R-q+r\cos P& -1& \cos P\\ p\cos Q+q\cos P-r& \cos P& -1\end{array}\right|$

$=\frac{1}{p}\left|\begin{array}{ccc}0& \cos R& \cos Q\\ 0& -1& \cos P\\ 0& \cos P& -1\end{array}\right|=0$

Using the projection formula:
$p=q\cos R+r\cos Q$