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## Dot Product of Vectors

The dot product (also known as the scalar product) is an operation on vectors that can tell you the angle between the vectors.

# Problem Solving

Consider the two vectors $$\vec{a}=(3,4,5)$$ and $$\vec{b}=(1,1,1).$$ What is the value of $$t$$ when the magnitude of $$\vec{a}+t\vec{b}$$ is minimum?

For what values of $$b$$ are the two vectors $$(-4, b)$$ and $$(b, {b}^{2} )$$ orthogonal?

If $$\vec{a}=(3,4,1)$$ and $$\vec{b}=(2,4,5),$$ what is the magnitude of the projection of $$\vec{a}$$ onto $$\vec{b}?$$

Consider a regular tetrahedron $$OABC$$ with edge length $$1.$$ $$P$$ is a point on $$\overline{OA}$$ such that $$\overrightarrow{OA}\cdot\overrightarrow{PB}=\frac{1}{3}$$ and $$Q$$ is a point on $$\overline{PB}$$ such that $$\overrightarrow{PB}\cdot\overrightarrow{QC}=0.$$ The vector $$\overrightarrow{QC}$$ can be expressed as $x\overrightarrow{OA}+y\overrightarrow{OB}+z\overrightarrow{OC}.$ Find the value of $$31(x+y+z).$$

Which of the following are correct where $$\vec{a}$$ and $$\vec{b}$$ are arbitrary vectors?

I. $$\lvert\vec{a}\cdot\vec{b}\rvert\le\lvert\vec{a}\rvert\lvert\vec{b}\rvert$$
II. $$\lvert\vec{a}+\vec{b}\rvert\le\lvert\vec{a}\rvert+\lvert\vec{b}\rvert$$
III. $$\lvert\vec{a}-\vec{b}\rvert\le\lvert\vec{a}\rvert-\lvert\vec{b}\rvert$$

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