Solve Vector Question: Find Shortest Distance from Point C to Line 1

  • Context: Undergrad 
  • Thread starter Thread starter padraig
  • Start date Start date
  • Tags Tags
    Vector
Click For Summary
SUMMARY

The discussion focuses on calculating the shortest distance from point C, represented by the position vector (2i - k), to line 1 defined by the equation r = (i + 2j - 3k) + m(4i - 5j - 3k). A key point made is that the line segment connecting point C to any point P on line 1 must be perpendicular to line 1 to achieve the shortest distance. Additionally, it is clarified that point C does not lie on line 2, which is irrelevant to the problem at hand.

PREREQUISITES
  • Understanding of vector equations and parametric representations of lines
  • Knowledge of vector calculus, specifically distance formulas between points and lines
  • Familiarity with the concept of orthogonality in vectors
  • Basic proficiency in manipulating vectors in three-dimensional space
NEXT STEPS
  • Study the formula for the distance from a point to a line in three-dimensional space
  • Learn how to derive the normal vector to a line from a given point
  • Explore vector projection techniques to find the shortest distance
  • Review examples of similar vector problems in calculus textbooks
USEFUL FOR

Students studying vector calculus, educators teaching geometry concepts, and anyone interested in solving problems related to distances in three-dimensional vector spaces.

padraig
Messages
10
Reaction score
0
Hi, I haven't done vectors in a while and just wondering if someone could refresh my memory in solving this question for me:

line 1: r = (i + 2j - 3k) + m(4i - 5j -3k)

line 2: r = (4i - 4j + 3k) + n(i - 2j + 2k)

Point C with pos. vector (2i - k) lies on line 2. Find the shortest distance from C to line 1

Cheers, ul probs find this no probs!

Pat
 
Physics news on Phys.org
the line segment joining C with a point P on line 1 must be normal on line 1 in order to yield the shortest distance
 
Check the wording of the problem again. It is fairly straight forward to find the distance from point C to line 1 (most calculus books have a formula) and line 2 is completely irrelevant- but C is NOT on line 2!
 

Similar threads

Find the shortest distance between the given vectors in 3D
  • · Replies 1 ·
Replies
1
Views
2K
A,B,C are 3 points with position vectors a,b,c Find length of median
  • · Replies 18 ·
Replies
18
Views
2K
Vector Geometry: Solving for Position Vectors and Equations in 3D
  • · Replies 11 ·
Replies
11
Views
5K
Vector of shortest distance between two skew lines
  • · Replies 11 ·
Replies
11
Views
4K
MHB Shortest distance between line and point
  • · Replies 6 ·
Replies
6
Views
3K
MHB Finding the Position Vector of Point $B$ on a Line Segment Between $P$ and $Q$
  • · Replies 2 ·
Replies
2
Views
2K
Finding the Shortest Distance from a Point to a Line in 3-Dimensional Space
  • · Replies 4 ·
Replies
4
Views
3K
MHB Smallest Distance $P(-8,4)$ to Line $y=6x$ at Origin
  • · Replies 2 ·
Replies
2
Views
2K
Min. distance from a line to a line
  • · Replies 5 ·
Replies
5
Views
2K
MHB [Linear Algebra] - Find the shortest distance d between two lines
  • · Replies 2 ·
Replies
2
Views
4K