Math Problem Statement
Solution
To find the distance from the point to the line given by the parametric equations: we can follow this procedure:
1. Vector Form of the Line:
The parametric equations describe a line. To write the line in vector form, we have: Thus, the line has direction vector and passes through the point .
2. Vector from Point on Line to Given Point:
Let be the vector from the point to the point on the line. This vector is:
3. Project onto the Line Direction:
To find the perpendicular distance, we need to project onto the direction vector . The formula for the projection of a vector onto another vector is: First, compute the dot products: Thus, the projection is:
4. Find the Perpendicular Vector:
The vector from to the line is the difference between and the projection:
5. Distance Calculation:
The distance from the point to the line is the magnitude of the perpendicular vector : Simplifying:
So, the distance from the point to the line is .
Would you like more details or have any questions?
Here are some related questions you might explore:
- How do you find the parametric equations of a line given two points?
- How is the formula for projecting one vector onto another derived?
- What are the different methods to find the distance between a point and a plane?
- How do you calculate the shortest distance between two skew lines?
- How do you determine if a point lies on a given line in 3D space?
Tip: Always verify if the projection vector lies in the same direction as the line, as this ensures you're calculating the correct perpendicular distance.
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Math Problem Analysis
Mathematical Concepts
Vector Projection
Distance from Point to Line in 3D
Parametric Equations
Formulas
Distance from a point to a line: d = |PQ x d| / |d|
Vector projection: proj_d PQ = (PQ ⋅ d / d ⋅ d) * d
Magnitude of a vector: |v| = sqrt(v1^2 + v2^2 + v3^2)
Theorems
Pythagorean Theorem
Vector Cross Product
Suitable Grade Level
Grades 11-12 or early college level
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