Mathematics-Online course: Vector Calculus-Scalar Product-Orthogonal Basis

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	Mathematics-Online course: Vector Calculus - Scalar Product
	Orthogonal Basis

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An orthonormal basis consists of

mutually orthogonal unit vectors $\vec{u}$ , $\vec{v}$ , $\vec{w}$ .

$\includegraphics[width=.7\linewidth]{a_on_basis}$

As illustrated in the figure, any vector $\vec{a}$ can be represented by a linear combination

$\displaystyle \vec{a} = (\vec{a}\cdot\vec{u})\vec{u} + (\vec{a}\cdot\vec{v})\vec{v} + (\vec{a}\cdot\vec{w})\vec{w}\,.$

The addends are the projections of $\vec{a}$ onto the axes generated by the vectors of the ONB. For the coefficients we have

$\displaystyle \left\vert\vec{a}\cdot\vec{u}\right\vert^2+\left\vert\vec{a}\cdot... ...t^2+\left\vert\vec{a}\cdot\vec{w}\right\vert^2=\left\vert\vec{a}\right\vert^2.$

In particular, we have

$\displaystyle \left(\begin{array}{c}a_1\\ a_2\\ a_3\end{array}\right) = a_1 \vec{e}_x + a_2 \vec{e}_y + a_3 \vec{e}_z$

for the canonical orthonormal basis of the Cartesian coordinate system

$\displaystyle \vec{e}_x = \left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),\qu... ...\right),\quad \vec{e}_z = \left(\begin{array}{c}0\\ 0\\ 1\end{array}\right).$

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automatically generated 10/30/2007