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

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In summary, the conversation discusses finding the shortest distance from a point C with a position vector (2i - k) to line 1, which is defined by the equation r = (i + 2j - 3k) + m(4i - 5j -3k). The conversation also mentions that line 2, defined by the equation r = (4i - 4j + 3k) + n(i - 2j + 2k), is irrelevant to finding the shortest distance. The shortest distance can be found by finding the line segment joining C with a point P on line 1 that is normal to line 1. Some resources, such as calculus books, may have a formula
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padraig
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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
 
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  • #2
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
 
  • #3
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!
 

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

How do I find the shortest distance from a point to a line using vectors?

The shortest distance from a point to a line can be found by using the formula: d = |(PC x n)| / |n|, where PC is the vector from the point to any point on the line, and n is the normal vector of the line. This formula can also be written as d = |(a x b)| / |b|, where a is the vector from the point to any point on the line and b is a vector parallel to the line.

Can I use any point on the line to calculate the shortest distance?

Yes, you can use any point on the line to calculate the shortest distance. The formula for the shortest distance does not depend on the specific point chosen on the line.

What if the point is not perpendicular to the line?

If the point is not perpendicular to the line, you can still use the formula d = |(PC x n)| / |n|, where PC is the vector from the point to any point on the line, and n is the normal vector of the line. This will give you the shortest distance from the point to the closest point on the line.

How do I find the normal vector of a line?

The normal vector of a line can be found by taking the cross product of any two vectors that are parallel to the line. This will give you a vector that is perpendicular to the line and can be used in the formula to find the shortest distance.

Can I use this method for finding the shortest distance between a point and a plane?

Yes, you can use a similar method for finding the shortest distance between a point and a plane. The formula for this is d = |(PC x n)| / |n|, where PC is the vector from the point to any point on the plane, and n is the normal vector of the plane. You can also use the formula d = |(a x b)| / |b|, where a is the vector from the point to any point on the plane and b is a vector parallel to the plane.

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