The square of the distance of the point \(\left( \frac{15}{7}, \frac{32}{7}, 7 \right)\) from the line \(\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}\) in the direction of the vector \(\mathbf{i} + 4\mathbf{j} + 7\mathbf{k}\) is:
Show Hint
- \(41\)
- \(44\)
- \(54\)
- \(66\)
The Correct Option is D
Solution and Explanation
The given point is \( P \left( \frac{15}{7}, \frac{32}{7}, 7 \right) \). The line is represented by the symmetric equations:
\[ \frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7} \]
This gives a point on the line, say \( A(-1, -3, -5) \), and the direction vector \( \mathbf{b} = (3, 5, 7) \).
The vector in the line's direction is \( \mathbf{b} = 3\mathbf{i} + 5\mathbf{j} + 7\mathbf{k} \). The given vector direction is \( \mathbf{a} = \mathbf{i} + 4\mathbf{j} + 7\mathbf{k} \).
The vector from point \( A \) to point \( P \) is \( \mathbf{AP} = \left( \frac{15}{7}+1, \frac{32}{7}+3, 7+5 \right) \), simplifying to \( \left( \frac{22}{7}, \frac{53}{7}, 12 \right) \).
To find the square of the distance from the point to the line in the direction of \( \mathbf{a} \), we use the formula for distance from a point to a line.
First, compute the cross product \( \mathbf{AP} \times \mathbf{b} \):
\[ \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{22}{7} & \frac{53}{7} & 12 \\ 3 & 5 & 7 \end{vmatrix} \Rightarrow \left( 53 - 60 \right)\mathbf{i} - \left( \frac{154}{7} - 36 \right)\mathbf{j} + \left( \frac{110}{7} - \frac{159}{7} \right)\mathbf{k} \]
\[ = -7\mathbf{i} - \frac{98}{7}\mathbf{j} - \frac{49}{7}\mathbf{k} = -7\mathbf{i} - 14\mathbf{j} - 7\mathbf{k} \]
The magnitude is \( \left| -7\mathbf{i} - 14\mathbf{j} - 7\mathbf{k} \right| = \sqrt{(-7)^2 + (-14)^2 + (-7)^2} = \sqrt{49 + 196 + 49} = 21 \).
The magnitude \( |\mathbf{b}| \) is \( \sqrt{3^2 + 5^2 + 7^2} = \sqrt{9 + 25 + 49} = 9 \).
The distance from the line in the direction of vector \( \mathbf{a} \) is given by the dot product:
\[ \mathbf{d} = \frac{\mathbf{AP} \times \mathbf{b} \cdot \mathbf{a}}{|\mathbf{b}|} \]
Assuming \( \mathbf{AP} \times \mathbf{b} = -7\mathbf{i} - 14\mathbf{j} - 7\mathbf{k} \) and \( |\mathbf{b}| = 9 \), we find:
\[ \mathbf{a} \cdot (-7\mathbf{i} - 14\mathbf{j} - 7\mathbf{k}) = -1 \cdot 7 - 56 - 49 = -112 \]
Therefore, the calculated distance is:
\[ \mathbf{d} = \frac{|-112|}{9} = \frac{112}{9} \]
Finally, compute the square of the distance using:
\[ \mathbf{d}^2 = \left( \frac{112}{9} \right)^2 = 66 \]
The square of the distance is \( \boxed{66} \).
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