How to project a vector onto an orthonormal basis

An orthonormal basis makes projection easier because the basis vectors are perpendicular and each has length 1. A dot product gives the component along each basis vector directly. We can add those components to get the projection onto their span.

Use one dot product for each basis direction

p = (x \cdot q_{1}) q_{1} + (x \cdot q_{2}) q_{2}

For an orthonormal basis with more vectors, continue the sum. If the basis vectors are the columns of $Q$ , the same formula is $p = Q Q^{T} x$ .

Find the two coefficients

Use $q_{1} = \frac{1}{\sqrt{2}} (1, 0, - 1)$ and $q_{2} = (\frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{1}{2})$ . Their lengths are 1 and their dot product is zero. For $x = (5, 6, - 1)$ ,

\begin{array}{l} x \cdot q_{1} = 3 \sqrt{2} \\ x \cdot q_{2} = 2 + 3 \sqrt{2} \end{array}

Combine the two components

Multiply each coefficient by its basis vector and add component by component. The projection has components

\begin{array}{l} p_{1} = 4 + \frac{3 \sqrt{2}}{2} \\ p_{2} = 3 + \sqrt{2} \\ p_{3} = - 2 + \frac{3 \sqrt{2}}{2} \end{array}

The projection lies in the plane spanned by the two basis vectors. The part left over is perpendicular to both of them.

Learn mathKrista KingAugust 11, 2021math, learn online, online course, online math

How to project a vector onto an orthonormal basis

Use one dot product for each basis direction

Find the two coefficients

Combine the two components

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