Geometry

Dot Product of Vectors

Dot Product - Problem Solving

         

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

by Brilliant Staff

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

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by Brilliant Staff

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

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by Brilliant Staff

Consider a regular tetrahedron OABCOABC with edge length 1.1. PP is a point on OA‾\overline{OA} such that OA→⋅PB→=13\overrightarrow{OA}\cdot\overrightarrow{PB}=\frac{1}{3} and QQ is a point on PB‾\overline{PB} such that PB→⋅QC→=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).

by Brilliant Staff

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

I. ∣a⃗⋅b⃗∣≤∣a⃗∣∣b⃗∣\lvert\vec{a}\cdot\vec{b}\rvert\le\lvert\vec{a}\rvert\lvert\vec{b}\rvert
II. ∣a⃗+b⃗∣≤∣a⃗∣+∣b⃗∣\lvert\vec{a}+\vec{b}\rvert\le\lvert\vec{a}\rvert+\lvert\vec{b}\rvert
III. ∣a⃗−b⃗∣≤∣a⃗∣−∣b⃗∣\lvert\vec{a}-\vec{b}\rvert\le\lvert\vec{a}\rvert-\lvert\vec{b}\rvert

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by Brilliant Staff
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