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48 Cards in this Set
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What is the distance formula in 3D? 


What is the formula for a sphere in 3D? 


What is a scalar? 
Is a quantity that has a magnitude describable by a real number and no direction. 

What is a vector? 
Is used to indicate a quanitity (such as displacement or velocity) that has both magnitude and direction. 

Dot Product 


Scalar projection of b onto a 


Scalar projection of a onto b 


Vector projection of b onto a 


Vector projection of a onto b 


Angle between vectors a and b of a dot product. 


When are vectors a and b called perpendicular or orthogonal? 
When the angle between them makes cosine zero such angles 0 deg, 180 deg, 360 deg, etc.
Two vectors a and b are orthogonal if and only if a dot b = 0. 

How to find the angle of a vector product? 


When are two nonzero vectors a and b parallel? 
If and only if a x b = 0 

What are the names of the angles between vector v?

The angle between vector v is:
 xaxis is alpha  yaxis is beta  zaxis is gamma 

What is the graph of a Line? 


What is the vector equation of a line? 
 r: The equation of the line.  r =  r0 =  v =  t = a parameter 

Equation of a Line 
Each t defines a point on the line. 

Unit Tangent Vector 


Magnitude (Length) of a Vector 


Ellipsoid Surface 


Cone Surface 


Elliptic Paraboloid Surface 


Hyperboloid of One Sheet Surface 


Hyperbolic Paraboloid Surface 


Hyperbolic of Two Sheets Surface 


Example of Vector Spaces 


What is a basis? 
A basis for a vector space is a set of vectors belonging to the vector space and that are linearly independent and that spans the vector space. 

What is a spanning set? 
A set of vectors is a spanning set of a vector space if every vector from the vector space can be written as a linear combination of the vectors from the spanning set. 

Equation of a plane 


Formula for the distance between a point and a plane 


Unit Normal Vector 


Unit Binormal Vector 


Curvature of a Curve 


Arc Length of a Curve 


Normal Plane 
Is a plane containing N(t) and B(t) and has T(t) as its normal vector. 

Osculating Plane 
Is a plane containing T(t) and N(t) and has B(t) as its normal vector. N(t) points to the center of the osculating circle. 

Rectifying Plane 
Is a plane containing T(t) and B(t) and has N(t) as its normal vector. 

Equation of a Tangent Plane for w=f(x,y,z) 


Change in f in the direction of vector u 


Maximum change in f 


Chain rule for the derivative of F with respect to s



Linear Approximation Equation 


Equation of a Tangent Plane for z=f(x,y) 


Volume of Parallelepiped 


Relative Minimum of a function with two variables 


Relative Maximum of a function with two variables 


Critical points of a function with two variables 


Second Derivative Test of a function with two variables 
