Dot, Cross, or Other
- Page ID
- 91736
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Deciding to Use the Dot Product, Cross Product, or Other Vector Arithmetic
You are working as a consultant and will be given a question that involves the use of the dot product, the cross product, or other vector arithmetic such as vector addition, subtraction or scalar multiplication. You must guide the client towards the best of the three choices so that the client can most efficiently get the job done.
A solar power roofing company wants to determine a vector perpendicular to the roof of a house and is given the coordinates of three non-colinear points on the roof, so they can easily find two vectors on the roof.
A chicken farmer wants to know the expected number of eggs each of the farm's chickens will lay per week. By definition if \(x_1,x_2,...,x_n\) are the possible outcomes of a discrete random variable and the outcomes have probabilities \(p_1,p_2,...,p_n\) of occurring, then the expected value is given by \(EV =\sum^{n}_{i=0}x_{i}p_{i}\).
An autonomous car programmer needs to track the location of the car that is on the road. In particular, the situation is that the car travels along a straight road for a certain distance and then turns onto another straight road for a different distance. The car's path can be represented by two vectors. The programmer wants to find the vector that goes from the car's starting point to the car's ending point.
A ski binding manufacturer needs to ensure that the binding releases the ski boot when 50% of the rotational force required to tear a ligament is applied to it due to a turn on the ski hill or a fall. The ski boot can be considered a vector and the force changing direction by gravity and the skier's muscles is another vector.
The municipal water agency will be transporting water up a hill and wants to know the total amount of work needed to pump it from the bottom of the hill to the top of the hill. The force of gravity is a vector pointing downwards and the bottom of the hill to the top of the hill is another vector.
An airplane engineer needs to know the total amount of external force that the plane is subject to. The two external forces are gravity that is pointing downwards and air resistance that is a combination of the wind and the direction and speed of the airplane.
A company that manufactures erasers needs to find the area of the side of the eraser which is in the shape of a parallelogram. The segment at the lower left corner to the upper left corner will form a vector as will the segment from the lower left corner to the lower right corner.
A computer video designer will use vector projections to translate three dimensional objects into two dimensions. With the viewing perspective from slightly above and in front of a person standing, the image shown on the computer screen will use the projection of the vector from the person's foot to head and will project it onto the vertical vector in front of the person.
The electric company is analyzing its poles to make sure that they are still parallel to each other. Using technology, they can get readings of the coordinates of the tops and bottoms of the poles and then want to use the easiest method to test if they are parallel.
Dot Cross Other
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