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Lyryx Assignment 6 (Question 6)
Name: Shaunak Dogra Date: 2024-11-10
Official Time: 14:39:15 Question 6 [10 points]
Let L be the line passing through the point P=(5, −1, −1) with direction vector →d=[0, 1, −2]T. Find the shortest distance d from the point P0=(−4, 4, 3) to L, and the point Q on L that is closest to P0. Use the square root symbol '√' where needed to give an exact value for your answer.
d =
√120
Q = (
0
,
0
,
0
)
Solution
To find the shortest distance from the point to the line , which passes through the point and has direction vector , we can use the following approach.
Step 1: Compute the vector
This vector goes from to :
Step 2: Find the projection of onto
The projection formula is:
-
Calculate :
-
Calculate :
-
So, [ \text{proj}_{\vec{d}} , \vec{PP_0} = \frac{-3}{5} \vec{d} = \frac{-3}{5} [0, 1, -2]^T = \left[ 0, -\frac{3}{5}, \frac{6}{5} \right] ]
Step 3: Find the perpendicular (shortest) distance vector
The perpendicular distance vector from to the line is: [ \vec{PP_0} - \text{proj}_{\vec{d}} , \vec{PP_0} = \left[ -9, 5, 4 \right] - \left[ 0, -\frac{3}{5}, \frac{6}{5} \right] = \left[ -9, \frac{28}{5}, \frac{14}{5} \right] ]
Step 4: Calculate the magnitude of the perpendicular distance vector
The distance is the magnitude of this vector:
So, the shortest distance is:
Step 5: Find the point on that is closest to
The point can be found by starting at and moving along by the scalar multiple found in the projection:
Final Answers
Would you like further explanation on any part of this solution?
- How do you determine the shortest distance from a point to a line in 3D?
- What role does the direction vector play in finding the point closest to ?
- Why is the projection operation used here?
- How would you find the distance if were a plane instead of a line?
- What is the geometric interpretation of the perpendicular distance vector?
Tip: When finding distances to lines or planes, visualizing projections can greatly simplify understanding the shortest paths.
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Math Problem Analysis
Mathematical Concepts
Analytic Geometry
Vectors
Distance in 3D Space
Formulas
Projection of vector: proj_d a = (a · d / d · d) * d
Distance between a point and a line in 3D
Theorems
Vector Projection Theorem
Pythagorean Theorem for distances in space
Suitable Grade Level
Grades 11-12
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