Waste less time on Facebook — follow Brilliant.
×

Shortest Distance from Origin - nD

This note is part of the set Shortest Distance from Origin

As work through the problems in this set, you might find a pattern in the formulae of distance of point from line, distance of point from plane and so on. In this note, we will prove that pattern.

Points in one dimensional space, lines in two dimensional space, and planes in there dimensional space are all examples of hyperplanes. A hyperplane in \( \mathbb{R}^{n}\) is an \( n -1\) dimensional figure, and divides \( \mathbb{R}^{n} \) in two halves. It is the set of all points that satisfy

\[a_{0} + a_{1} x_{1} + a_{2} x_{2} + a_{3} x_{3} + \cdots + a_{n} x_{n} = 0 \]

for some real numbers \(a_{i}\). For any given point \( (b_{1}, b_{2}, b_{3},\ldots, b_{n}) \), prove that the shortest distance of a hyperplane in \(\mathbb{R}^{n} \) from that point is given by

\[ d = \frac{|a_{0} + a_{1} b_{1} + a_{2} b_{2} + a_{3}b_{3} + \cdots + a_{n}b_{n}|}{\sqrt{a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + \cdots +a_{n}^{2} }} \]

Note by Pranshu Gaba
1 year, 8 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

Let \(\vec{N} = (a_1,a_2,a_3,a_4,\cdots, a_n) \) be a vector in \(\mathbb{R}^n\).

Let \(\vec{P} = (x_1,x_2,\cdots, x_n) \) be a point in \(\mathbb{R}^n\) such that \(\vec{P}\cdot\vec{N} \) is always constant. Let this constant be \(k (k \in R) \).

Hence, the locus of point \(P\) is given by:

\(\displaystyle \Rightarrow \vec{P} \cdot \vec{N} = k \)

\(\displaystyle \Rightarrow \displaystyle \sum_{i=1}^{n}a_i x_i = k \)

The above equation represents a hyperplane in \(\mathbb{R}^n\) and \(\vec{N} \) is the vector normal to the hyperplane.

Comparing it with the equation of given hyperplane yields \(k= - a_0\).

Now consider \(\vec{B} = (b_1,b_2,\cdots, b_n) \).

The distance of point \(B\) from the hyperplane will be measured along the unit vector normal to the hyperplane.

To measure this distance, we find the extent of the join of \(B\) and a point \(R\) on the hyperplane, along the unit vector normal to the hyperplane. Hence,

\(Distance = \left| \vec{BR} \cdot \hat{N} \right| \)

\(Distance = \left| \frac{ (\vec{R} - \vec{B}) \cdot \vec{N}} {\left| \vec{N} \right| } \right|\)

\(Distance = \frac{\left| \vec{R} \cdot \vec{N} - \vec{B} \cdot \vec{N} \right|} {\left| \vec{N}\right|} \)

\(Distance = \frac{\left| k - \displaystyle \sum_{i=1}^n a_i b_i \right|}{\sqrt{\displaystyle \sum_{j=1}^n a_j^2}} \)

\(Distance = \frac{\left| a_1b_1 + a_2b_2 + a_3b_3 + \cdots + a_nb_n - k\right|}{\sqrt{a_1^2 + a_2^2 + \cdots + a_n^2}} \)

That is for a general hyperplane. For the given hyperplane, \(k=-a_0\).

\(\displaystyle \Rightarrow Distance = \frac{\left|a_0 + a_1b_1 + a_2b_2 + \cdots + a_nb_n \right|}{\sqrt{a_1^2 + a_2^2 + \cdots + a_n^2}} \)

Harsh Khatri - 1 year, 8 months ago

Log in to reply

That's correct. Well done Harsh!

Pranshu Gaba - 1 year, 8 months ago

Log in to reply

Thank you!

Harsh Khatri - 1 year, 8 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...