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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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