Orthogonal Complement Calculator

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

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

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

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.

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