‌L1:2x+3y+z=4 ‌x+2y+z=5 ∴ line L1 in standard form is ‌
x+7
1
=‌
y−6
−1
=‌
z
1
and equation of line L2 is: ‌
x−2
1
=‌
y+1
−1
=‌
z−3
1
Equation of plane M:2x+y−2z=6. Let coordinate of Q=(λ+2,−λ−1,λ+3). ∵Q lies on plane M ∴λ=−9 ∴ coordinate of Q=(−7,8,−6). For foot of perpendicular R(x1,y1,z1) ‌
x1−2
2
=‌
y1+1
1
=‌
z1−3
−2
=‌
−(4−1−6−6)
9
∴ Coordinate of R=(4,0,1)
‌∴PQ=9√3‌ units ‌ ‌QR=√234‌ units ‌ ‌PR=3‌ units ‌ Let θ be acute angle between PQ and PR, then cos‌θ=‌