Closest point on a line to a given point in 3D space

[mspaic]

Hey guys, I have this assingment in algebra oveer the internet. I get as many tries as i need for a question, as long as i get it right by tomorrow midnight.

This is the ONLY one i got left so if anyone can help i'd really appreciate it...

------------

Using the system of linear equations below, answer the following questions.

2x1 + 5x2 + 7x3= -2

-6x1 - 15x2 -21x3= 6

a) (5 marks) Find the solution of the given system of linear equations, Q, that is closest to the point P(0, -7, 0).

b) (5 marks) What is the distance between the point P(0, -7, 0) and the point Q, from above?

------------------

ok here are the answers, these are 100% sure right but i don't know how the program go them...

The correct answers to parts a) and b) are:

Q(0.846, -4.885, 2.962)
distance = 3.737

 

[mspaic]

Also, i think i know how to do part B.
I just don't know how to find point Q.

 

[Parth Dave]

Well, you'll note that the two lines are actually the same thing (they are scaler multiples of each other). That being the question basically asks you to find the shortest distance between the line and that point P. Also, geometrically speaking, the equation you have given represents a plane. My suggestion would be to create a line that is perpendicular to your plane and then find the point of intersection, which will be Q. Then find the magnitude of the vector QP for your distance.

 

[mspaic]

That is a very good point. I now see examples in the book with the same thing.
HOWEVER, i still have a problem finding the equation of the line that is perpendicular to the plane. If i remember from high school, since the line is perpendicular to the plane, it's direction vector and the diretion vectors of the plane should be perpendicular. Therefore, the direction verctor of the line should be PARALLEL to the NORMAL of the plane (2,5,7). Is that correct? also how do i find the point i need to write the equation of the line. thanks a lot

 

[Parth Dave]

Yea that is correct, the direction vector of the line will be the normal of the plane. And remember, you want the point of intersection of that line (which passes through P) and the plane.

 

[mspaic]

thank you thank you thank you i finally got the question lol
now i don't have to run around tomorrow around the univeristy gettin help YAA