Find the distance of the point $(2, 12, 5)$ from the point of intersection of the line $\to r = 2^i - 4^j + 2^k + λ (3^i + 4^j + 2^k)$ and the plane $\to r \cdot (^i - 2^j +^k) = 0.$

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Solution

General point on the line:
$x = 2 + 3 λ, y = - 4 + 4 λ, z = 2 + 2 λ$
The equation of the plane:
$\to r . (^i - 2^j +^k) = 0$
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 $λ$ .
$[(2 + 3 λ)^i + (- 4 + 4 λ)^j + (2 + 2 λ)^k] . (^i - 2^j +^k) = 0$
$[(2 + 3 λ) .1 + (- 4 + 4 λ) (- 2) + (2 + 2 λ) .1] = 0$
$12 - 3 λ = 0,$ or $λ = 4$
Therefore the point of intersection is:
$(2 + 3 (4), - 4 + 4 (4), 2 + 2 (4)) = (14, 12, 10)$
Distance of this point from $(2, 12, 5)$ is
$= \sqrt{(14 - 2)^{2} + (12 - 12)^{2} + (10 - 5)^{2}}$
$= \sqrt{12^{2} + 5^{2}}$
$= 13$

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