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What is the Foot of the Perpendicular from a Point to a Plane?
Grade Level:
Class 12
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Definition
What is it?
The 'Foot of the Perpendicular from a Point to a Plane' is simply the point on the plane that is closest to the given point. Imagine dropping a straight line from the point, hitting the plane at a perfect 90-degree angle; where that line touches the plane is the 'foot'. It's the unique point where the perpendicular line from the point meets the plane.
Simple Example
Quick Example
Imagine you are standing on the ground (the plane) and holding a cricket ball (the point) directly above your head. If you drop the ball, it will fall straight down and hit the ground at a specific spot. That spot where the ball lands on the ground is the 'foot of the perpendicular' from the ball's initial position to the ground.
Worked Example
Step-by-Step
Let's find the foot of the perpendicular from point P(1, 2, 3) to the plane x + 2y - z = 5.
1. The line passing through P(1, 2, 3) and perpendicular to the plane x + 2y - z = 5 will have direction ratios equal to the normal vector of the plane. The normal vector is (1, 2, -1).
---2. So, the equation of the line is (x - 1)/1 = (y - 2)/2 = (z - 3)/(-1) = lambda (let's say 'k').
---3. Any point on this line can be written as (k + 1, 2k + 2, -k + 3).
---4. The foot of the perpendicular, let's call it Q, is the point where this line intersects the plane. So, Q must satisfy the plane's equation: (k + 1) + 2(2k + 2) - (-k + 3) = 5.
---5. Simplify the equation: k + 1 + 4k + 4 + k - 3 = 5. This gives 6k + 2 = 5.
---6. Solve for k: 6k = 3, so k = 1/2.
---7. Substitute k = 1/2 back into the point Q's coordinates: Q(1/2 + 1, 2(1/2) + 2, -1/2 + 3).
---8. Calculate the coordinates: Q(3/2, 1 + 2, -1/2 + 6/2) = Q(1.5, 3, 2.5).
Answer: The foot of the perpendicular is (1.5, 3, 2.5).
Why It Matters
Understanding the foot of the perpendicular is crucial in fields like robotics and computer graphics to calculate the shortest distance and position objects accurately. It helps engineers in designing structures and even scientists in mapping geographical data, ensuring precision in complex systems.
Common Mistakes
MISTAKE: Using the point's coordinates as the direction ratios for the perpendicular line. | CORRECTION: The direction ratios of the perpendicular line are the coefficients of x, y, and z from the plane's equation (which form the normal vector).
MISTAKE: Forgetting to substitute the parametric form of the line (with 'k' or 'lambda') back into the plane's equation to find the value of the parameter. | CORRECTION: Always substitute the general point on the line into the plane's equation to find the specific 'k' that makes the point lie on the plane.
MISTAKE: Incorrectly simplifying the algebraic equation for 'k' after substitution. | CORRECTION: Double-check all additions, subtractions, and multiplications when solving for the parameter 'k' to avoid calculation errors.
Practice Questions
Try It Yourself
QUESTION: Find the foot of the perpendicular from the point (0, 0, 0) to the plane 2x + 3y - z = 4. | ANSWER: (8/14, 12/14, -4/14) or (4/7, 6/7, -2/7)
QUESTION: A mobile tower is at point (2, 5, 1). The ground is represented by the plane x - y + 2z = 7. Find the coordinates of the point on the ground directly below the tower. | ANSWER: (4, 3, 5)
QUESTION: Find the coordinates of the foot of the perpendicular from the point A(3, -1, 11) to the line (x/2) = (y - 2)/3 = (z - 3)/4. (Hint: First find the plane containing A and perpendicular to the line). | ANSWER: (2, 5, 7)
MCQ
Quick Quiz
If a point P(x1, y1, z1) is given and a plane Ax + By + Cz + D = 0, what does the foot of the perpendicular represent?
The point on the plane farthest from P
The point where the line parallel to the plane passes through P
The point on the plane closest to P
The origin of the coordinate system
The Correct Answer Is:
C
The foot of the perpendicular is defined as the point on the plane that lies on the perpendicular line dropped from the given point, making it the closest point on the plane to the given point.
Real World Connection
In the Real World
In urban planning, when designing a new flyover (a point in space) over a road (a plane), engineers need to find the exact point on the road directly below the flyover for pillar placement. This is exactly finding the foot of the perpendicular, ensuring stable and safe construction.
Key Vocabulary
Key Terms
PERPENDICULAR: Forming a right angle (90 degrees) | PLANE: A flat, two-dimensional surface extending infinitely | NORMAL VECTOR: A vector perpendicular to a plane | PARAMETRIC EQUATION: An equation that defines a set of quantities as functions of one or more independent variables called parameters.
What's Next
What to Learn Next
Great job understanding this! Next, you can explore how to find the shortest distance from a point to a plane. This concept builds directly on finding the foot of the perpendicular, as the distance is simply the length of the line segment connecting the point to its foot on the plane. Keep learning!
