Let $A(1,-1,2), B(6,11,2), C(1,2,6)$ be three points. If $l_1, \mathrm{~m}_1, \mathrm{n}_1$ are the…
Let $A(1,-1,2), B(6,11,2), C(1,2,6)$ be three points. If $l_1, \mathrm{~m}_1, \mathrm{n}_1$ are the direction cosines of $\mathrm{AB}$ and $l_2, \mathrm{~m}_2, \mathrm{n}_2$ are the direction cosines of $\mathrm{AC}$, then $\left|l_1 L_2+\mathrm{m}_1 \mathrm{~m}_2+\mathrm{n}_1 \mathrm{n}_2\right|=$
$\frac{63}{65}$
$\frac{36}{65}$
$\frac{16}{65}$
$\frac{13}{64}$
Solution
Direction ratios of $A B$ are
$a_1=6-1=5, b_1=11-(-1)=12, c_1=2-2=0$
Direction ratio's of $A C$ are
$\begin{aligned} & a_2=1-1=0, b_2=2-(-1)=3, c_2=6-2=4 \\ & \therefore \quad l_1=\frac{a_1}{\sqrt{a_1^2+b_1^2+c_1^2}}=\frac{5}{\sqrt{25+144+0}}=\frac{5}{13} \\ & m_1=\frac{b_1}{\sqrt{a_1^2+b_1^2+c_1^2}}=\frac{12}{\sqrt{25+144+0}}=\frac{12}{13} \\ & n_1=\frac{c_1}{\sqrt{a_1^2+b_1^2+c_1^2}}=\frac{0}{\sqrt{25+144+0}}=0 \\ & l_2=\frac{a_2}{\sqrt{a_2^2+b_2^2+c_2^2}}=\frac{0}{\sqrt{0+9+16}}=0 \\ & m_2=\frac{b_2}{\sqrt{a_2^2+b_2^2+c_2^2}}=\frac{3}{\sqrt{0+9+16}}=\frac{3}{5}\end{aligned}$
$\begin{aligned} & n_2=\frac{c_2}{\sqrt{a_2^2+b_2^2+c_2^2}}=\frac{4}{\sqrt{0+9+16}}=\frac{4}{5} \\ & \text { Then, }\left|l_1 l_2+m_1 m_2+n_1 n_2\right| \\ & =\left|\frac{5}{13} \times 0+\frac{12}{13} \times \frac{3}{5}+0 \times \frac{4}{5}\right|=\frac{36}{65}\end{aligned}$