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.
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