A line L passing through the point (-1, 2, -3) is perpendicular to the plane P given by $2x + 3y + z + 5 = 0$.
\(3x + 3 = 2y - 4 = 6z + 18\)
To find the equation of line \( L \) that is perpendicular to the given plane and passes through the point \((-1, 2, -3)\), we need to follow these steps:
- Understanding Perpendicularity: A line perpendicular to a plane will have a direction vector that is parallel to the normal vector of the plane. The equation of the plane is \(2x + 3y + z + 5 = 0\). The normal vector to this plane is \(\mathbf{n} = \langle 2, 3, 1 \rangle\).
- Point on the Line: The line \( L \) passes through the point \((-1, 2, -3)\). Therefore, the line can be represented parametrically as:
- \(x = -1 + 2t\)
- \(y = 2 + 3t\)
- \(z = -3 + t\)
- Eliminating the Parameter: To form symmetric equations, eliminate \( t \):
- From \( x = -1 + 2t \), we rearrange to get \( t = \frac{x + 1}{2} \).
- From \( y = 2 + 3t \), we rearrange to get \( t = \frac{y - 2}{3} \).
- From \( z = -3 + t \), we rearrange to get \( t = z + 3 \).
- Matching with Options: The symmetric form matches with:
\(- (x + 1) = y - 2 = z + 3\)
Thus, the correct answer is that the equation of line \( L \) is
\(- (x + 1) = y - 2 = z + 3\)
Similar Questions
- Which of the following are the direction ratios of S?
- If \(p\) is the perpendicular distance from origin to the plane passing through \((1, 0, 0)\), \((0, 1, 0)\) and \((0, 0, 1)\), then what is \(3p^2\) equal to ?
- What are the direction ratios of a line M parallel to the plane P ?
- Which of the following are the direction ratios of the line of intersection of the given planes?
- What is the equation of the plane \(P\)?
Important Questions from Equation in a Plane
- Which of the following are the direction ratios of S?
- If \(p\) is the perpendicular distance from origin to the plane passing through \((1, 0, 0)\), \((0, 1, 0)\) and \((0, 0, 1)\), then what is \(3p^2\) equal to ?
- What are the direction ratios of a line M parallel to the plane P ?
A straight line segment is 36 cm long. Points are to be marked on the line from both the end points. From each end, the first point is at a distance of 1 cm from the end, the second point is at a distance of 2 cm from the first point and the third point is at a distance of 3 cm from the second point and so on.
If the points on the ends are not counted and the common points are counted as one, what is the number of points?
- Which of the following are the direction ratios of the line of intersection of the given planes?
