Orthogonal Complement Calculator

Find the basis of an orthogonal complement step by step

This calculator will find the basis of the orthogonal complement of the subspace spanned by the given vectors, with steps shown.

A
$$$\mathbf{\vec{v_{1}}}$$$ $$$\mathbf{\vec{v_{2}}}$$$

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please contact us.

Your Input

Find the orthogonal complement of the subspace spanned by $$$\mathbf{\vec{v_{1}}} = \left[\begin{array}{c}1\\2\\3\end{array}\right]$$$, $$$\mathbf{\vec{v_{2}}} = \left[\begin{array}{c}4\\1\\7\end{array}\right]$$$.

Solution

Since every vector in the orthogonal complement should be orthogonal to every vector in the given subspace, we need to find the null space of $$$\left[\begin{array}{ccc}1 & 2 & 3\\4 & 1 & 7\end{array}\right]$$$.

The basis for the null space is $$$\left\{\left[\begin{array}{c}- \frac{11}{7}\\- \frac{5}{7}\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).

This is the basis for the orthogonal complement.

Answer

The basis for the orthogonal complement is $$$\left\{\left[\begin{array}{c}- \frac{11}{7}\\- \frac{5}{7}\\1\end{array}\right]\right\}\approx \left\{\left[\begin{array}{c}-1.571428571428571\\-0.714285714285714\\1\end{array}\right]\right\}.$$$A