Scalar Product
Scalar Product
\[\mathbf a\cdot\mathbf b=|\mathbf a||\mathbf b|\cos\theta=a_1b_1+a_2b_2+a_3b_3,\qquad 0\leq\theta\leq\pi.\]
Scalar projection
| Consequence | Condition |
|---|---|
| Perpendicularity | \(\mathbf a\cdot\mathbf b=0\) for nonzero \(\mathbf a,\mathbf b\) |
| Angle | \(\displaystyle\cos\theta=\frac{\mathbf a\cdot\mathbf b}{|\mathbf a||\mathbf b|}\) for nonzero vectors |
| Self-product | \(\mathbf a\cdot\mathbf a=|\mathbf a|^2\) |
| Distribution | \(\mathbf a\cdot(\mathbf b+\mathbf c)=\mathbf a\cdot\mathbf b+\mathbf a\cdot\mathbf c\) |
The scalar product is commutative and distributive, but it is not an associative vector product. Cancellation is not generally valid.
A scalar product is a number. In components, multiply matching entries and add; in geometric form, multiply the two magnitudes and the cosine of their angle. For nonzero vectors with \(0\le\theta\le\pi\), a positive scalar product means \(0\le\theta<\frac\pi2\), zero means \(\theta=\frac\pi2\), and a negative value means \(\frac\pi2<\theta\le\pi\).
Expand \(|\mathbf a\pm\mathbf b|^2\) as a self-product: \(|\mathbf a\pm\mathbf b|^2=|\mathbf a|^2\pm2\mathbf a\cdot\mathbf b+|\mathbf b|^2\). Take the non-negative square root only after simplifying. In general, \(|\mathbf a+\mathbf b|\ne|\mathbf a|+|\mathbf b|\).
For nonzero vectors, equality \(|\mathbf a\cdot\mathbf b|=|\mathbf a||\mathbf b|\) tests parallelism. For an angle between two directed vectors retain the sign of the dot product; for the smaller angle between two unoriented lines use its absolute value.
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