Dot Product of Vectors

Dot Product - Problem Solving


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

For what values of bb are the two vectors (4,b) (-4, b) and (b,b2) (b, {b}^{2} ) orthogonal?

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

Consider a regular tetrahedron OABCOABC with edge length 1.1. PP is a point on OA\overline{OA} such that OAPB=13\overrightarrow{OA}\cdot\overrightarrow{PB}=\frac{1}{3} and QQ is a point on PB\overline{PB} such that PBQC=0.\overrightarrow{PB}\cdot\overrightarrow{QC}=0. The vector QC\overrightarrow{QC} can be expressed as xOA+yOB+zOC.x\overrightarrow{OA}+y\overrightarrow{OB}+z\overrightarrow{OC}. Find the value of 31(x+y+z).31(x+y+z).

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

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


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