Question

The length of the perpendicular drawn from the point \((1, 2, 3)\) to the line \((X - 6)/3 = (y - 7)/2 = (Z - 7)/-2\) is:

• A. \(4 \, \text{units}\)
• B. \(5 \, \text{units}\)
• C. \(6 \, \text{units}\)
• D. \(7 \, \text{units}\)

Hint:

To calculate the distance of a point from a line, use the vector cross-product and magnitude formulas.

Answer 1

The Correct Option is D

The equation of the line can be written in vector form as: \[ \mathbf{r} = (6, 7, 7) + t(3, 2, -2). \] Let the point \((1, 2, 3)\) be denoted as \(\mathbf{P}\) and the position vector of any point on the line as \(\mathbf{r_0}\). The direction vector of the line is \(\mathbf{d} = (3, 2, -2)\). The perpendicular distance from a point \((x_1, y_1, z_1)\) to a line is given by: \[ D = \frac{\|\mathbf{d} \times (\mathbf{r_0} - \mathbf{P})\|}{\|\mathbf{d}\|}. \] Step 1: Compute \((\mathbf{r_0} - \mathbf{P})\): \[ \mathbf{r_0} - \mathbf{P} = (6 - 1, 7 - 2, 7 - 3) = (5, 5, 4). \] Step 2: Compute the cross product \(\mathbf{d} \times (\mathbf{r_0} - \mathbf{P})\): \[ \mathbf{d} \times (\mathbf{r_0} - \mathbf{P}) = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k}
3 & 2 & -2
5 & 5 & 4 \end{vmatrix}. \] \[ \mathbf{d} \times (\mathbf{r_0} - \mathbf{P}) = \mathbf{i}(2 \cdot 4 - (-2) \cdot 5) - \mathbf{j}(3 \cdot 4 - (-2) \cdot 5) + \mathbf{k}(3 \cdot 5 - 2 \cdot 5). \] \[ \mathbf{d} \times (\mathbf{r_0} - \mathbf{P}) = \mathbf{i}(8 + 10) - \mathbf{j}(12 + 10) + \mathbf{k}(15 - 10). \] \[ \mathbf{d} \times (\mathbf{r_0} - \mathbf{P}) = (18, -22, 5). \] Step 3: Compute the magnitude of the cross product: \[ \|\mathbf{d} \times (\mathbf{r_0} - \mathbf{P})\| = \sqrt{18^2 + (-22)^2 + 5^2} = \sqrt{324 + 484 + 25} = \sqrt{833}. \] Step 4: Compute the magnitude of the direction vector \(\mathbf{d}\): \[ \|\mathbf{d}\| = \sqrt{3^2 + 2^2 + (-2)^2} = \sqrt{9 + 4 + 4} = \sqrt{17}. \] Step 5: Compute the distance \(D\): \[ D = \frac{\|\mathbf{d} \times (\mathbf{r_0} - \mathbf{P})\|}{\|\mathbf{d}\|} = \frac{\sqrt{833}}{\sqrt{17}} = \sqrt{\frac{833}{17}} = 7. \]