Assumption of variable
a=\[a_1\] b =\[b_1\] c =\[c_1\]
d=\[a_2\] k=\[b_2\] f=\[c_2\]
If \[a_1\] x+\[b_1\]y+\[c_1\]z =\[d_1\] and and \[a_2\] x+\[b_2\]y+\[c_1\]z =\[d_2\] are straight lines then \[a_1\] \[b_1\] \[c_1\] and \[a_2\] \[b_2\] \[c_2\] are called direction rations of the lines.
Acute angle between lines is given by cosx= \[ \frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{(a_1^2+b_1^2+c_1^2)(a_2^2+b_2^2+c_2^2)}}\]
Example- Find the angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.
solution
\[a_1\]= 4 \[b_1\]=-3 \[c_1\]=5
\[a_2\]=3 \[b_2\]=4 \[c_2\]=5
cosx= \[ \frac{4(3)+(-3)4+55}{\sqrt{(4^2+(-3)^2+5^2)(3^2+4^2+5^2)}}\]
cosx=\[\frac{1}{2}\] hence x= 60 degrees or x= \[\frac{\Pi }{3}\]
