Question

The perpendicular distance from the point \( (-1,1,0) \) to the line joining the points \( (0,2,4) \) and \( (3,0,1) \) is:

• A. \( 10 \)
• B. \( \frac{2\sqrt{5}}{5} \)
• C. \( \frac{5}{\sqrt{2}} \)
• D. \( 8 \)

Hint:

For a point to line distance in 3D, use: \( D = \frac{|(\mathbf{r_0} - \mathbf{r_1}) \cdot (\mathbf{d} \times \mathbf{p})|}{|\mathbf{d} \times \mathbf{p}|}. \)

Answer 1

The Correct Option is C

\documentclass{article} \usepackage{amsmath} \usepackage{amssymb} \begin{document} Solution:
We are given: - Point \( P(-1, 1, 0) \) - Line passing through points \( A(0, 2, 4) \) and \( B(3, 0, 1) \) We need to find the perpendicular distance from point \(P\) to the line \( \overline{AB} \). Step 1: Direction Vector of the Line
The direction vector of the line joining points \(A\) and \(B\) is: \[ \vec{AB} = (3 - 0)\hat{i} + (0 - 2)\hat{j} + (1 - 4)\hat{k} = 3\hat{i} - 2\hat{j} - 3\hat{k} \] Step 2: Vector \( \vec{AP} \)
\[ \vec{AP} = (-1 - 0)\hat{i} + (1 - 2)\hat{j} + (0 - 4)\hat{k} = -\hat{i} - \hat{j} - 4\hat{k} \] Step 3: Perpendicular Distance Formula
The perpendicular distance from point \( P \) to the line passing through \( A \) in the direction of \( \vec{AB} \) is given by: \[ d = \frac{|\vec{AP} \times \vec{AB}|}{|\vec{AB}|} \] Step 4: Compute the Cross Product \( \vec{AP} \times \vec{AB} \)
\[ \vec{AP} \times \vec{AB} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
-1 & -1 & -4
3 & -2 & -3 \end{vmatrix} \] \[ = \hat{i} \left((-1)(-3) - (-4)(-2)\right) - \hat{j} \left((-1)(-3) - (-4)(3)\right) + \hat{k} \left((-1)(-2) - (-1)(3)\right) \] \[ = \hat{i} (3 - 8) - \hat{j} (3 + 12) + \hat{k} (2 + 3) \] \[ = -5\hat{i} - 15\hat{j} + 5\hat{k} \] Step 5: Compute Magnitudes
\[ |\vec{AP} \times \vec{AB}| = \sqrt{(-5)^2 + (-15)^2 + 5^2} = \sqrt{25 + 225 + 25} = \sqrt{275} = 5\sqrt{11} \] \[ |\vec{AB}| = \sqrt{(3)^2 + (-2)^2 + (-3)^2} = \sqrt{9 + 4 + 9} = \sqrt{22} \] Step 6: Compute the Distance
\[ d = \frac{5\sqrt{11}}{\sqrt{22}} = \frac{5}{\sqrt{2}} \] Step 7: Final Answer
\[ \boxed{\frac{5}{\sqrt{2}}} \] Final Answer: (C) \( \frac{5}{\sqrt{2}} \)