Vector calculator
- Calculate the norm of a vector
- Calculate orthogonal vector of a given vector (2D plan)
- Normalize a vector
Euclidean Norm of a vector
The Euclidean norm of a vector `\vecu` of coordinates (x, y) in the 2-dimensional Euclidean space, can be defined as its length (or magnitude) and is calculated as follows :
`norm(vecu) = sqrt(x^2+y^2)`
The norm (or length) of a vector `\vecu` of coordinates (x, y, z) in the 3-dimensional Euclidean space is defined by:
`norm(vecu) = sqrt(x^2+y^2+z^2)`
Example: Calculate the norm of vector `[[3],[2]]`
`norm(vecu) = sqrt(3^2+2^2) = sqrt(13)`
Orthogonal Vector
Definition
Two vectors of the n-dimensional Euclidean space are orthogonal if and only if their dot product is zero.
The following propositions are equivalent :
- `\vecu _|_ \vecv`
- Vectors `\vecu` and `\vecv` are orthogonal
- Their dot product is zero: `\vecu . \vecv = 0`
How to calculate the orthogonal vector ?
Let `\vecu` be a vector of coordinates (a, b) in the Euclidean plane `\mathbb{R^2}`. Any `\vecv` vector of coordinates (x, y) satisfying this equation is orthogonal to `\vecu`:
`\vecu . \vecv = 0`
`a.x + b.y = 0`
If `b != 0` then `y = -a*x/b`
Therefore, all vectors of coordinates `(x, -a*x/b)` are orthogonal to vector `(a,b)` whatever x. We note that all these vectors are collinear (have the same direction).
For x = 1, we have `\vecv = (1,-a/b)` is an orthogonal vector to `\vecu`.
Vector Normalization
Definition : Let `\vecu` be a non-zero vector. The normalized vector of `\vecu` is a vector that has the same direction than `\vecu` and has a norm which is equal to 1.
We note `\vecv` the normalized vector of `\vecu`, then we have,
`\vecv = \vecu/norm(vecu)`
Example : Normalization of the vector of coordinates (3, -4) in the Euclidean plane
We compute its norm,
`\norm(vecu) = sqrt(3^2 + (-4)^2) = sqrt(25) = 5`
The normalized vector of `\vecu` is therefore `\vecv = \vecu/norm(vecu) = (3/5 , -4/5)`