Physics 135 - Lecture 1B - 5.3.18.pptx - Physics for the...

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Physics ToolsPhysics for the Life SciencesLecture#1BStewart Hall
More Vector OperationsWe know how to add and subtract vectors, either geometrically or by componentsWhat does it mean tomultiply two vectors?
The Scalar ProductWhen we do the “scalar product” we multiply two vectors to obtain a quantity which is a pure scalarThis is also called the “dot product” or “inner product”We combine two things with magnitude and direction to get just a magnitude…rθFFThe scalar product r F is definedr F = |r| |F| cos(θ) (magnitude of r times magnitude of F times thecosine of the angle in between)rDot product of r and F
What does this do for us?
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Fθr|| = r cos(θ)FθF|| = F cos(θ)Note that the dot product is commutative: r F = F r = r F cos(θ)Scalar Product Propertiesr F = Fr cos(θ) = F [r cos(θ)] = ±F r||rr F = Fr cos(θ) = r [F cos(θ)] = ±r F||r
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How could we use this?Imagine you take a trip consisting of four different components, and you just want to know “how far east did I go”?Multiply each vector by a unit vector in the East” direction and add them up…PQRSQRSDistance East ! Eˆ ! Eˆ ! Eˆ ! EˆP!!!!P xˆ Q xˆ R xˆ S xˆy(N)x(E)
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The VectorProductA second operation: two vectors produce a new vector, so the vector productCombine two vectors to get a new vectorThis is also called the ‘crossproduct’This measures how much they’re perpendicularrFThe vector product r x F is defined as a new vector with magnitude |r x F| = |r| |F| sin(θ)=r F sin(θ) and with a direction given by the Right-Hand Rule (in this case into the slide…)Cross product of r and FrFθ

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