# Collinearity of vectors in 2d space

This online calculator determines if 2d space vectors are collinear

This online calculator can find collinear 2d vectors in a given set of vectors. Enter vector coordinates x and y, separated by space, one line per vector. The calculator will find if any of them are collinear. You can find the description of the method with formulas below the calculator

#### Collinearity of vectors in 2d space

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### How to find if vectors are collinear.

Collinear vectors are the vectors parallel to one line or lying on one line. Such vectors are linearly dependent, and you can express one vector through another as

$\vec{a}=\lambda \vec{b}$

Hence the most obvious way to check if two 2d vectors with coordinates ${x_1;y_1}, {x_2;y_2}$ are collinear is to check the equality

$\frac{x_1}{x_2}=\frac{y_1}{y_2}$

However, this can't be applied if any of the coordinates is zero.

Happily, there is another, more elegant criteria of vectors collinearity: The determinant of the matrix of vector coordinates should be equal to zero

$det \left| \begin{matrix} x_1&x_2\\y_1&y_2}\end{matrix} \right| = 0$

or

$x_1y_2-y_1x_2=0$

Why? If determinant is equal to zero, hence the matrix rank is 1, which means that only one column in the matrix is independent, while another one is linearly dependent on it, which is essentially the condition of collinearity of vectors.

This expression is used by the calculator to check vectors for collinearity. Vectors are taken in pairs, and each set of collinear vectors is placed on the separate line in the result list.

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