Dot and Cross Product

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Projection notation

"Projection of b onto a" "Projection of b along a"

How do you find the angle between two vectors?

*use dot product geometric formula!

What is the dot product used for?

- Find the cosθ - Find if an angle is acute, obtuse, or orthogonal - Find work - Projection of vectors

What is the cross product used for?

- Find the sinθ - Find a mutually perpendicular vector - Prove that two vectors are parallel - Find the area of a parallelogram or triangle created by two vectors

Notation of a vector

<a,b,c) ai + bj + ck

Create a vector given points

A = (a1, a2, a3) B = (b1, b2, b3) (b1-a1,b2-a2,b3-a3)

What does a dot product result in?

A SCALAR!

What does a cross product result in?

A VECTOR!

(a•b)xc

MEANINGLESS

(axb)•c

SCALAR

Volume Equation

V = |a•(bxc)| *a,b,c must be vectors *CAN BE IN ANY ORDER!

(axb)xc

VECTOR

k(axb)+c

VECTOR

Normalize a vector

Vector/magnitude *vector with a length of 1 *if you want to create a vector with a specific length/magnitude, normalize the vector and multiply it by the desired length/magnitude

When are vectors coplanar?

WHEN THE VOLUME = 0! |a•(bxc)| = 0

When is the angle between two vectors acute?

When a•b=+

When is the angle between two vectors obtuse?

When a•b=-

When is the angle between two vectors orthogonal?

When a•b=0

When are vectors parallel?

When the vectors are scalar multiples! Ex: <2,3,4> and <4,6,8> OR CROSS PRODUCT IS 0!

Dot product

a•b = (a)(d) + (b)(e) + (c)(f) <a,b,c> <d,e,f>

How do you create a vector between two points?

ending point - initial point

What are the components of i, j, and k?

i: <1,0,0> j: <0,1,0> k: <0,0,1>

What is the right hand rule with unit vectors?

ixj=k jxk=i kxi=j jxi=-k kxj=-i ixk=-j

How do you multiply a vector by a constant?

k<a,b,c> = <ak,bk,ck>

Work done by a force

work = f • w f = force vector (weight) w = distance vector (length)

How do you find the area of a triangle?

|axb|/2=(|a||b|sinθ)/2

How do you find the area of a parallelogram?

|axb|=|a||b|sinθ


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