Dot Product of Vectors
If \(\vec{u} = \langle 1, 2, 3 \rangle\) and \(\vec{v} = \langle 4, 5, 6\rangle\), then what is \(\vec{u} \cdot \vec{v}\)?
In the figure above, the point \(H\) is the perpendicular foot from point \(A\) on \(\overrightarrow{OB}.\) If \(\lvert\overline{OA}\rvert=7,\lvert\overline{OB}\rvert=9,\) and \(\lvert\overline{OH}\rvert=5,\) what is \(\overrightarrow{OA}\cdot\overrightarrow{OB}?\)
Note: The figure is not drawn to scale.
If \(\vec{u}, \vec{v},\) and \(\vec{w}\) are vectors, then which of the following is not a valid operation?
\[ \begin{align*} \text{A. } & \hspace{.35cm} (\vec{u}\cdot\vec{v}) + |\vec{w}| \\ \text{B. } & \hspace{.35cm} \vec{u}\cdot\vec{v}\cdot\vec{w} \\ \text{C. } & \hspace{.35cm} \vec{u}\cdot(\vec{v} - \vec{w}) \end{align*}\]
Remember! Both inputs for a dot product must be vectors.
If the vectors \(\vec{v}=(\sqrt{3},4,\sqrt{2})\) and \(\vec{u}=(\sqrt{6},0,k)\) are perpendicular, what must be the value of \(k?\)
Hint: If two vectors are perpendicular, then their dot product is 0.
If \(\vec{a}\cdot\vec{b}=3, \vec{a}\cdot\vec{c}=4, \vec{b}\cdot\vec{c}=-2,\) and \(\lvert\vec{b}\rvert=1,\) then evaluate \[(\vec{a}-\vec{b})\cdot(2\vec{b}+3\vec{c})\]