Geometry

Dot Product of Vectors

Dot Product - Properties

         

If u=1,2,3\vec{u} = \langle 1, 2, 3 \rangle and v=4,5,6\vec{v} = \langle 4, 5, 6\rangle, then what is uv\vec{u} \cdot \vec{v}?

In the figure above, the point HH is the perpendicular foot from point AA on OB.\overrightarrow{OB}. If OA=7,OB=9,\lvert\overline{OA}\rvert=7,\lvert\overline{OB}\rvert=9, and OH=5,\lvert\overline{OH}\rvert=5, what is OAOB?\overrightarrow{OA}\cdot\overrightarrow{OB}?

Note: The figure is not drawn to scale.

If u,v,\vec{u}, \vec{v}, and w\vec{w} are vectors, then which of the following is not a valid operation?

A. (uv)+wB. uvwC. u(vw) \begin{aligned} \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{aligned}

Remember! Both inputs for a dot product must be vectors.

If the vectors v=(3,4,2)\vec{v}=(\sqrt{3},4,\sqrt{2}) and u=(6,0,k)\vec{u}=(\sqrt{6},0,k) are perpendicular, what must be the value of k?k?

Hint: If two vectors are perpendicular, then their dot product is 0.

If ab=3,ac=4,bc=2,\vec{a}\cdot\vec{b}=3, \vec{a}\cdot\vec{c}=4, \vec{b}\cdot\vec{c}=-2, and b=1,\lvert\vec{b}\rvert=1, then evaluate (ab)(2b+3c)(\vec{a}-\vec{b})\cdot(2\vec{b}+3\vec{c})

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