General point on the line: x = 2 + 3λ, y = -8 + 4λ, z = 2 + 2λ
The equation of the plane: →r ⋅ (^i - 6^j + ^k) = 0
The point of intersection of the line and the plane: Substituting general point of the line in the equation of plane and finding the particular value of λ.
[(2 + 3λ)^i + (-8 + 4λ)^j + (2 + 2λ)^k] ⋅ (^i - 6^j + ^k) = 0
[(2 + 3λ).1 + (-8 + 4λ)(-6) + (2 + 2λ).1] = 0
2 + 3λ + 48 - 24λ + 2 + 2λ = 0
52 - 19λ = 0
19λ = 52
λ = 52/19
Therefore the point of intersection is:
(2 + 3(52/19), -8 + 4(52/19), 2 + 2(52/19))
= (2 + 156/19, -8 + 208/19, 2 + 104/19)
= (192/19, 40/19, 142/19)
Distance of this point from (2, 12, 5) is
= √[(192/19 - 2)^2 + (40/19 - 12)^2 + (142/19 - 5)^2]
= √[(154/19)^2 + (-198/19)^2 + (37/19)^2]
= √[23716/361 + 39204/361 + 1369/361]
= √(64289/361)
= 13.37