Distance between a point and a line (vectors) (KristaKingMath)

Learn how to use vectors to find the distance between a point and a line, given the coordinate point and parametric equations of the line. Use the parametric equations to find a vector that gives direction numbers and a coordinate point. Find a vector between the two coordinate points. Then take the cross product of the two vectors, and the magnitude of the cross product. Use a distance formula to find the distance between the point and the line.

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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)

Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”

[XetXetable]

For those wondering where the formula comes from, it comes from the fact that, if we have two vectors, u and v, the area of the parallelogram they form will be | u x v |.

If we have a line passing through some point, O, and another point not on our line, P, we can form two vectors. Firstly, d, the direction vector along the line, and OP, the vector from O to P. They will form a parallelogram who's height is the distance between the line and the point.

The height of a parallelogram is just the area divided by the base-length. The base-length of our parallelogram is | d |. This gives us the final formula;

| d x OP |

    | d |

An alternative, maybe more intuitive, formula is given by the vector orthogonal to the projection. This gives the same answer as the above formula;

Orth   d
      ᴾᴼ

[someguy]

The formula at 4:30 was all I needed to solve and understand this problem. Not sure what the other 8 and a half minutes of this video covered, but that formula was genius! (using cross product to find the area of a parallelogram drawn using the line and point and then dividing out the base of the parallelogram, which leaves you the height, which is what you were solving for)

[muzaffarmustafa]

your concept is far way better than my lecture notes, super short, sweet and straight to the point and of course the answer :) thank you!

[Drewtaku]

Where did you get this problem? because on my assignment I literally have the exact same values... the math gods favor me this day!

[trucommander]

You are absolutely amazing! Seriously! After 5 years...I am still coming to you! You are a wonderful soul!

[cosmasmwamwembe]

Thanks, you explained it better in 8 mins than an hour lecture

[amibiggestfan]

Thank you so much for the step by step explanation!! It's so much more easier to understand and it helped me through my tutorials!

[Diamonddrake]

You neglected to tell us WHY you use that formula. It's some kind of vector projection . But I don't remember which, or how you know to use that formula. Which I what I was watching the video for. :(

[theodorlee]

Thank's so very much for going step by step in your videos! You've helped me a lot! I like how you teach us by breaking down the parts in the formula into definitions...Sometimes I have to pause and replay a few parts because your notation is unorthodox :P

[solomonmule]

I've noticed that when deriving the vector B, you subtracted <4, 1, -2> by <1, 3, 4>. Is there any particular reason that you did it in this order, and would it be OK if I did it as <1, 3, 4> - <4, 1, -2>?

[zekethefreak]

Your videos help me so much with having a better understanding in my Calc class

[shyamikaachintha]

Thank you very muchhhh... the steps are clearly explained in the video ❤❤👍

[kevinpg]

I appreciate your time in this! thank you!

[ABKidrocks]

Can you explain how you got the formula at 4:30

[rexclone]

A much faster approach would be through perpendicular projection of vector b on vector a, with that calculating the closest point on the line to the point of the line, and then finishing by calculating the distance between the two points.

[solomonmule]

I've noticed that when deriving the vector B, you subtracted <4, 1, -2> by <1, 3, 4>. Is there any particular reason that you did it in this order, and would it be OK if I did it as <1, 3, 4> - <4, 1, -2>?please someone to make it clear

[candyrapper]

Thank you for making this video! God bless you and your family!

[zacherylouis]

Thank you so much! This makes so much more sense and wasn't explained in my textbook.

[ExploreSphere]

i have a confusion. i think the formula is not correct according to the recent research it should be bxa instead of axb bcz axb!=bxa.
if you think i'm wrong then explain me plz

[CAth]

Is this the shortest distance between the point and the line?