Let the direction cosines of line of intersection of the planes x−y=0 and 2x+y+z=0 are a1,b1,c1 so a1−b1=0 and 2a1+b1+c1=0 By cross-multiplication method, we have
‌
a1
−1−0
=‌
−b1
1−0
=‌
c1
1+2
⇒‌
a1
1
=‌
b1
1
=‌
c1
−3
Similarly, let the direction cosines of line of intersection of the planes 2x−z=0 and x+y−3z=0 are a2,b2,c2 so, 2a2−b2=0 a2+b2−3c2=0 By cross-multiplication method, we have
‌
a2
3−0
=‌
−b2
−6−0
=‌
c2
2+1
⇒‌
a2
1
=‌
b2
2
=‌
c2
1
So angle ′θ′ between lines having directions cosines a1,b1,c1 and a2,b2,c2 respectively . is