Vectors Flashcards
Vectors
Directional arrows that describe an objects motion and magnitude
Naming vectors
[OriginPoint][EndPoint]
Line over head
ie PQ
Scalar-vector multiplication
Multiplying the vector magnitude by a constant (c)
Endpoint (x,y,z) becomes
(cx,cy,cz)
Scalar multiple
A vector whose magnitude is multiplied by a contant
Parallel vectors
Vectors that are scalar multiples of eachother
Equal vectors
Vectors of the same direction and magnetude
Not necessarily the same position
Zero vector
A vector component with no length
Vector addition
u+v
Reposition the vectors ‘u’ and ‘v’ so that they form two sides of a triangle
Use c^2=a^2+b^2, one at a time for multiple vectors
Add endpoint vx to ux, vy to uy, and vz to uz
Vector subtraction
u-v
Change the direction of vector ‘v’
Reposition the vectors ‘u’ and new ‘v’ so that they form two sides of a triangle
Use c^2=a^2+b^2
Subtract endpoint vx from ux, vy from uy, and vz from uz
Position vectors
Vector with its head at (0,0)
Standard position
(0,0)
The general starting point for object motion
X-component
The vector that measures how far the object moves in simply the x-direction, if y and z were ignored
Y-component
The vector that measures how far the object moves in simply the y-direction, if x and z were ignored
Z-component
The vector that measures how far the object moves in simply the z-direction, if y and x were ignored
Vector magnetude
The ‘length’ of the vector, typically a representation of acceleration
Denoted |PQ|
Use Pythagorean theorem
|PQ|=√[(x-xi)^2+(y-yi)^2+(z-zi)^2]
Writing vector coordinates
Where u1is the origin point for the vector and u2 is the vector head
Vector head
End-point of the vector
Origin point
Point at which the vector begins
Unit vector
Any vector with a magnitude of 1
Right hand coordinate system
Graph ‘z’ up, ‘y’ right, and ‘x’ diagonal towards you
Vector magnetude 3D coordinate plane
r=√[(x-xi)^2+(y-yi)^2+(z-zi)^2]
Dot product
Product of the magnitudes of each vector and the ‘cos’ of the angle between the two vectors, is equal to the sum of their component products
u⊙v=|u||v|cosθ=(ux)(vx)+(uy)(vy)+(uz)(vz)
Orthogonal vectors
Vectors whose dot product is zero
u⊙v=0
Yields perpendicular vectors
Projections
Proj=|u|cosθ(v/|v|)
Also equals
v[(u⊙v)/(v⊙v)]
Work
Dot product of the force vector and the traveling vector (in which the object actually travels, gives the angle measure if you have the magnitudes
W=F⊙d=|F||d|cosθ
Cross product
|u x v|=|u||v|sinθ
Matrix representation of cross product
i. j. k.
u. = u x v =|j and k|i+|i and k|j+|j and i|k
v
Torque
Cross product of force and radius, outward pressure due to force on a plane, think of tightening a wrench (torque is the force pushing a screw in or out of a hole)
r x F
Vector-valued functions
Expressing an algebraic curve in space with three functions i,j,k
v(t)=[f(t)]i+[g(t)j+[h(t)]k
Limits of vector-valued functions
Lim t→a v(t)=[f(t)]i+[g(t)j+[h(t)]k
Tangent vector
Basically the derivative if the vector valued function
When v(t)=[f(t)]i+[g(t)j+[h(t)]k,
Then
v’(t)= [f’(t)]i + [g’(t)j + [h’(t)]k
Unit Tangent Vector
Gives the direction but not the length of the tangent vector at point-t T(t)=v'(t)/|v'(t)| For v(t)= [f(t)]i + [g(t)j + [h(t)]k
Integral of a vector valued function
When v(t)=[f(t)]i+[g(t)j+[h(t)]k, ∫v(t)= [∫f(t)]i + [∫g(t)j + [∫h(t)]k
Uniform motion
Motion in a straight line
Arc Length for Vector functions
L=∫√[f’(t)^2+g’(t)^2+h’(t)^2]dt = ∫|v’(t)|dt
Curvature
A measure of how quickly the direction of the curve changes over a given interval
Given as K=|dT/ds|
Where T= ∂v/|∂v|
And ‘s’ equals arc length over a set interval
Velocity given a curve of a constant radius
Always orthogonal
r⊙v=0
Curvature Formula
K=|v” x v’|/(|v|^3)
Principal Unit Normal Vector
Given as N=(dT/ds)/(|dT/ds|)
Where T= ∂v/|∂v|
Acceleration Components
a=(a1)N+(a2)T
Equation for a plane through a vector and point
d=ax+by+cz
Parallel planes
Their normal vectors are scalar multiples of eachother
Orthogonal plane
Normal vectors of which are zero
Cylinder
A surface containing of all lines parallel to eachother
Trace
Set of points at which a surface intersects a plane parallel to either x,y, or z coordinate planes
Xy trace
A trace on a plane paralel to z-axis
Xz-trace
A trace on a plane parallel to y-axis
Yz-trace
A trace on a plane parallel to x-axis
Basic Ellipsoid formula
(x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=1
a- ±x-vertices
b- ±y-vertices
c- ±z-vertices
Elliptic parabola
(x^2)/(a^2)+(y^2)/(b^2)=z
a- ±x-vertices
b- ±y-vertices
c- ±z-vertices
One-sheet Hyperboliod
(x^2)/(a^2)+(y^2)/(b^2)-(z^2)/(c^2)=1
a- ±x-vertices
b- ±y-vertices
c- ±z-vertices
Two-sheet hyperboliod
-(x^2)/(a^2)-(y^2)/(b^2)+(z^2)/(c^2)=1
a- ±x-vertices
b- ±y-vertices
c- ±z-vertices
Elliptic Code
(x^2)/(a^2)+(y^2)/(b^2)=(z^2)/(c^2)
a- ±x-vertices
b- ±y-vertices
c- ±z-vertices
Hyperbolic paraboloid
(x^2)/(a^2)-(y^2)/(b^2)=z
a- ±x-vertices
b- ±y-vertices
c- ±z-vertices
Equations of a line through a point at a specific vector
(X,y,z)=(z0,y0,x0)+t(a,b,c)
Normal vector
Vector that determines the orientation of a plane
Limit of a Vector-Valued Function
Lim r(t)=L Where r(t) is the vector valued function
Vectors may represent
Velocity, momentum, acceleration, force, or coordinates
Converting velocity vectors to momentum vectors
Multiply each i j k component by mass
Converting acceleration vectors to momentum vectors
Calculate the indefinate integral ∫a(t)+Vo=v(t) for each of the i j k directions then multiply by mass to find each component-function p(t)=m*v(t)
Converting force vectors to momentum vectors
Divide each force i j k component by object mass to find acceleration components. Calculate the indefinate integral ∫a(t)+Vo=v(t) for each of the i j k directions then multiply by mass to find each component-function p(t)=m*v(t)
Converting directional coordinate functions to momentum vectors
Take the derivative ∂s(t)=v for each i j k component, multiply by object mass to find p=m*v in each component direction
Converting momentum vectors to velocity vectors
Divide the momentum vector for each of the i j k components by object mass
Converting coordinate functions to velocity vectors
Take the derivative ∂s(t)=v for each i j k component,
Converting force vectors into velocity function vectors
Divide each force i j k component by object mass to find acceleration components then calculate the indefinate integral ∫a(t)+Vo=v(t) for each of the i j k directions
Converting acceleration vectors into velocity function vectors
Calculate the indefinate integral ∫a(t)+Vo=v(t) for each of the i j k directions
Converting velocity vectors to coordinate functions of time
Calculate the indefinate integral ∫v(t)+So=s(t) for each of the i j k directions
Converting acceleration vectors to coordinate functions of time
Calculate the indefinate integral ∫a(t)+Vo=v(t) and then ∫v(t)+So=s(t) for each of the i j k directions
Converting momentum vectors to coordinate functions of time
Divide each of the i j k components by object mass then calculate the indefinate integral ∫v(t)+So=s(t) for each of the x y z directions
Converting force vectors to coordinate functions of time
Divide each i j k component by mass for acceleration vectors then calculate the indefinate integral ∫a(t)+Vo=v(t) and then ∫v(t)+So=s(t) for each of the x y z directions
Converting force vectors to acceleration vectors
Divide each vector component magnitude by object mass
Converting velocity function vectors to acceleration vectors
Take the derivative ∂v(t)=a for each i j k component,
Converting momentum function vectors to acceleration vectors
Divide each i j k component by object mass then take the derivative ∂v(t)=a for each i j k component
Converting coordinate function vectors to acceleration vectors
Take the derivative ∂s(t)=v for each i j k component, then take the derivative ∂v(t)=a for each i j k component,
Converting acceleration vectors to force vectors
Multiply by mass for each of the ijk vector components
Converting velocity functions to force vectors
Take the derivative ∂v(t)=a and multiply by object mass for each i j k component
Converting momentum functions to force vectors
Divide each i j k component by object mass then take the derivative ∂v(t)=a and multiply by mass for each i j k component
Converting coordinate functions to force vectors
Take the derivative ∂s(t)=v for each i j k component, then take the derivative ∂v(t)=a and multiply by mass for each i j k component,
Scalar multiplication
The magnetude of the vector is changed by multiplying a constant to all directional components
Vector addition
If you look at the two vectors as the legs of a triangle (changing their position but not their diection or magnetude)- tail to head, then the final vector would be the third leg of that triangle
Vector subtraction
If you look at the two vectors as the legs of a triangle (changing their position but not their diection or magnetude)- tail to tail, then the final vector would be the third leg of that triangle
Cross product
When the two vectors are written one on top of the other in a matrix
The determinant of the vertical-submatrix for each direction
Always just a vector, of moment for those two vectors
Dot product
When the two vectors are written one on top of the other in a matrix
The sum of the determinants of the vertical-submatrix for each direction
Always just a number
Scalar triple product
The dot product of a vector and the cross product of two others
Order of vector opperstions
Cross product always comes before dot product
Vector triple product
The cross product of a vector and the cross product of two other vectors
Vector Gradient
Sum of the partial derivatives in each of those vector field directions
Measures the rate and direction of change in a scalar field, points to greatest potential, where the tangent is zero - lowest point for gravitational potential, highest voltage point for electrical-field potential, most positively charged point for magnetic field potential; where the ball/electron will most likely ‘roll’ (accelerate)
Vector curl
Describes a change in vector direction in terms of its projection onto the other tangent lines
Look it up and practice
Vector divergence
Describes the 3ddirection inwhich an object will tend to move
Limit of the Double integral of a projection
Variables in vectors
(x,y,z)=(i,j,k)
