Question

The distance of the point $P (4,6,-2)$ from the line passing through the point $(-3,2,3)$ and parallel to a line with direction ratios $3,3,-1$ is equal to :

• A. $2 \sqrt{3}$
• B. $\sqrt{14}$
• C. 3
• D. $\sqrt{6}$

Answer 1

The Correct Option is B

Equation of line is \(\frac{x+3}{3} = \frac{y-2}{3} = \frac{z-3}{-1} = \lambda\)
\(M(3λ−3,3λ+2,3−λ) \)
D.R of PM\(( 3λ−7,3λ−4,5−λ) \)
Since PM is perpendicular to line 
\(⇒3(3λ−7)+3(3λ−4)−1(5−λ)=0 \)
\(⇒λ=2 \)
\(⇒M(3,8,1)⇒PM=\) \(\sqrt{14}​\)

Answer 2

1. The parametric equation of the line passing through \((-3, 2, 3)\) and parallel to direction ratios \((3, 3, -1)\) is: \[ \mathbf{r} = (-3 + 3\lambda, 2 + 3\lambda, 3 - \lambda), \] where \(\lambda\) is a scalar. 2. The coordinates of P are \( (4, 6, -2) \). 3. The vector from \((-3, 2, 3)\) to \(P(4, 6, -2)\) is: \[ \mathbf{b} = (4 - (-3), 6 - 2, -2 - 3) = (7, 4, -5). \] 4. Direction vector of the line is: \[ \mathbf{a} = (3, 3, -1). \] 5. The perpendicular distance \(d\) from a point to a line is given by: \[ d = \frac{|\mathbf{a} \times \mathbf{b}|}{|\mathbf{a}|}. \] 6. Compute \(\mathbf{a} \times \mathbf{b}\): \[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k}
3 & 3 & -1
7 & 4 & -5 \end{vmatrix} = \mathbf{i} \begin{vmatrix} 3 & -1
4 & -5 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 3 & -1
7 & -5 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 3 & 3
7 & 4 \end{vmatrix}. \] \[ \mathbf{a} \times \mathbf{b} = \mathbf{i}(-15 + 4) - \mathbf{j}(-15 + 7) + \mathbf{k}(12 - 21), \] \[ \mathbf{a} \times \mathbf{b} = (-11, 8, -9). \] 7. Magnitude of \(\mathbf{a} \times \mathbf{b}\): \[ |\mathbf{a} \times \mathbf{b}| = \sqrt{(-11)^2 + 8^2 + (-9)^2} = \sqrt{121 + 64 + 81} = \sqrt{266}. \] 8. Magnitude of \(\mathbf{a}\): \[ |\mathbf{a}| = \sqrt{3^2 + 3^2 + (-1)^2} = \sqrt{9 + 9 + 1} = \sqrt{19}. \] 9. Distance \(d\): \[ d = \frac{\sqrt{266}}{\sqrt{19}} = \sqrt{\frac{266}{19}} = \sqrt{14}. \] The perpendicular distance of a point from a line in 3D space involves the cross product of the direction vector and the vector from the point to the line. The formula ensures the shortest distance between the point and the line.