Find the distance of the point (-1,-51-10) from the point of intersection of the line $\vec{r} = 2 \hat{i} - \hat{j} + 2 \hat{k} + λ (3 \hat{i} + 4 \hat{j} + 2 \hat{k})$ and the plane $\vec{r} . (\hat{i} - \hat{j} + \hat{k}) = 5$

Solution

Cartesian equation of line and plane,

$\frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 2}{2} : (Line) x - y + z - 5 = 0 : (Plane) Let Q (α, β, γ) be point of intesection of line and plane which will satisfy both equation \frac{α - 2}{3} = \frac{β + 1}{4} = \frac{γ - 2}{2} = λ α = 3 λ + 2 β = 4 λ - 1 γ = 2 λ + 2 Also, α - β + γ - 5 = 0 3 λ + 2 - 4 λ + 1 + 2 λ + 2 - 5 = 0 λ = 0 ∴ α = 2, β = - 1, γ = 2 \Rightarrow Q \equiv (2, - 1, 2) l (PQ) = \sqrt{(- 1 - 2)^{2} + (- 5 + 1)^{2} + (- 10 - 2)^{2}} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13 units$

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Three Dimensional Geometry

Find the distance of the point (-1,-51-10) from the point of intersection of the line $\vec{r} = 2 \hat{i} - \hat{j} + 2 \hat{k} + λ (3 \hat{i} + 4 \hat{j} + 2 \hat{k})$ and the plane $\vec{r} . (\hat{i} - \hat{j} + \hat{k}) = 5$

Some More Questions From Three Dimensional Geometry Chapter

If α, β, γ are the angles which a line makes with the axes, prove that sin²α + sin² β + sin²γ = 2.

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Three Dimensional Geometry

Find the distance of the point (-1,-51-10) from the point of intersection of the line r→ = 2i^ -j^ + 2k^ + λ (3i^ + 4j^ + 2k^) and the plane r→. (i^-j^ + k^) = 5

Some More Questions From Three Dimensional Geometry Chapter

If α, β, γ are the angles which a line makes with the axes, prove that sin2α + sin2 β + sin2γ = 2.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Find the distance of the point (-1,-51-10) from the point of intersection of the line $\vec{r} = 2 \hat{i} - \hat{j} + 2 \hat{k} + λ (3 \hat{i} + 4 \hat{j} + 2 \hat{k})$ and the plane $\vec{r} . (\hat{i} - \hat{j} + \hat{k}) = 5$

If α, β, γ are the angles which a line makes with the axes, prove that sin²α + sin² β + sin²γ = 2.