Dot and Cross Product
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
