A line is inclined at equal angles to all three (x-, y- and z- axes) and passed through the origin. A second line passes through (1,2,4) and (0,0,1). What is the angle between the two lines?

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Answer 1 · 2018-05-22T17:58:00

# arccos(sqrt42/7)#.

Suppose that, the line #L_1# is inclined, to all #3# Axes, at

an angle #alpha#.

This means that the direction cosines of #L_1# are

#(cosalpha,cosalpha,cosalpha)#.

In other words, the direction #vec(l_1)# of #L_1" is "(1,1,1)#.

Similarly, for the second line #L_2," since, "(1,2,4) & (0,0,1) in L_2#,

the direction #vec(l_2)=(1,2,4)-(0,0,1)=(1,2,3)#.

So, the reqd. #/_theta# between #L_1 and L_2#,by definition, is

the #/_" between "vec(l_1) and vec(l_2)#.

#:. costheta={vec(l_1)*vec(l_2)}/(||vec(l_1)||||vec(l_2)||}#.

Here, #vec(l_1)*vec(l_2)=(1,1,1)*(1,2,3)=1+2+3=6#,

#||vec(l_1)||=sqrt(1^1+1^2+1^2)=sqrt3#,

#||vec(l_2)||=sqrt(1^1+2^2+3^2)=sqrt14#.

#:. costheta=6/(sqrt3*sqrt14)=6/sqrt42=sqrt42/7#.

#:. theta=arccos(sqrt42/7)#.