Dot Product of Vectors

Dot Product - 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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