Scalar Product Tests (HL only)

\(\mathbf a\cdot\mathbf b=a_1b_1+a_2b_2+a_3b_3,\qquad\mathbf a\cdot\mathbf a=|\mathbf a|^2\)

For nonzero vectors, \(\cos\theta=\dfrac{\mathbf a\cdot\mathbf b}{|\mathbf a||\mathbf b|}\), where \(0\le\theta\le\pi\). A negative scalar product gives \(\frac\pi2<\theta\le\pi\), including antiparallel vectors.

\(\mathbf a\perp\mathbf b\iff\mathbf a\cdot\mathbf b=0\), for nonzero vectors. Parallel vectors satisfy \(\mathbf a=k\mathbf b\), equivalently \(|\mathbf a\cdot\mathbf b|=|\mathbf a||\mathbf b|\).

\((k\mathbf a)\cdot\mathbf b=k(\mathbf a\cdot\mathbf b),\quad \mathbf a\cdot(\mathbf b+\mathbf c)=\mathbf a\cdot\mathbf b+\mathbf a\cdot\mathbf c\).

Example

For \(\mathbf a=(1,2,-1)\), \(\mathbf b=(3,-1,1)\), \(\mathbf a\cdot\mathbf b=3-2-1=0\); the nonzero vectors are perpendicular.