Term
What is the difference between a vector and a scalar? 

Definition
vector = had magnitude and direction
scalar = has only magnitude 


Term
How do you add two vectors geometrically? 

Definition
by putting them end to end such that one starts where the other ends 


Term
How do you add two vectors algebraically? 

Definition
by finding the distance between the start point of the first and the end point of the second using the distance formula 


Term
If a is a vector and c is a scalar, how is ca related to a geometrically? 

Definition
ca has the same direction as a but has a magnitude that is c times greater or smaller than a 


Term
If a is a vector and c is a scalar, how do you find ca algebraically? 

Definition
multiply the components of vector a each by the scalar c 


Term
How do you find the vector from one point to another? 

Definition
if the points are P_{1}= (x_{0},y_{0},z_{0}) and P_{2}=(x_{1},y_{1},z_{1}) then the vector v=< x_{1}x_{0 }, y_{1}y_{0 },z_{1}z_{0 }>



Term
How do you find the dot product of a • b of two vectors if you know their lengths and the angle between them? 

Definition


Term
How do you find the dot product of a • b of two vectors if you know their components? 

Definition
if a = < a_{1}, b_{1}, c_{1}> and b = <a_{2}, b_{2}, c_{2}>
a • b = a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} 


Term
What is the expression for the scalar projection of b onto a? 

Definition


Term
What is the expression for the vector projection of b onto a?


Definition
proj_{a}b = (a • b)/a^{2}*a
or
proj_{a}b = comp_{a}b * a/a 


Term
How do you find the cross product a x b of two vectors if you know their lengths and the angle between them? 

Definition


Term
How do you find the cross product a x b of two vectors if you know their components? 

Definition
if a = < a_{1}, b_{1}, c_{1}> and b = <a_{2}, b_{2}, c_{2}>
a x b = < b_{1}c_{2 } c_{1}b_{2 }, c_{1}a_{2 } a_{1}c_{2} , b_{1}a_{2}  a_{1}b_{2} > 


Term
How do you find the area of the parallelogram determined by a and b? 

Definition


Term
How do you find the volume of the parallelepiped determined by a, b, and c? 

Definition
V = scalar triple product = a • (b x c) 


Term
How do you find a vector perpendicular to a plane? 

Definition
find the normal vector, which is the cross product of two vectors lying on the plane 


Term
How do you find the angle between two intersecting planes? 

Definition
by finding the dot or cross product of the intersecting planes normal vectors and solving using the magnitude angle relationship 


Term
How do you write the vector equation for a line? 

Definition
r = r_{0} + tv
where r_{0} is a point on the line and v is a vector parallel to the line and the line depends on the scalar t 


Term
How do you write parametric equations for a line? 

Definition
v = < a, b, c>
r_{0} = < x_{0}, y_{0}, z_{0}>
x = x_{0} + ta
y = y_{0} + tb
z = z_{0} + tc 


Term
How do you write symmetric equations for a line? 

Definition
v = < a, b, c>
r_{0} = < x_{0}, y_{0}, z_{0}>
(x  x_{0})/a = (y  y_{0})/b = (z  z_{0})/c



Term
How do you write vector equations for a plane? 

Definition


Term
How do you write scalar equations for a plane? 

Definition
n = < a, b, c >
r = < x, y, z >
r_{0} = < x_{0}, y_{0}, z_{0} >
a(x  x_{0}) + b(y  y_{0}) + c(z  z_{0}) = 0



Term
How do you tell if two vectors are parallel? 

Definition
if the cross product is zero 


Term
How do you tell if two vectors are perpendicular? 

Definition
if the dot product is zero 


Term
How do you tell if two planes are parallel? 

Definition
if the cross product of their normal vectors is zero 


Term
How do you find the distance from a point to a line? 

Definition
find the magnitude of the proj_{a}b if a is a vector on the line and b is a vector from a point on the line to the point 


Term
How do you find the distance from a point to a plane? 

Definition
use the distance formula
plane: 0 = ax + by + cz + d
point: (x_{0}, y_{0}, z_{0})
D = ax_{0} + by_{0} + cz_{0 }+ d/√(a^{2 }+ b^{2} + c^{2}) 


Term
How do you find the distance between two lines? 

Definition
find the cross product of two vectors, one that lies on each plance
using that vector and any vector that goes from one line to the other, find the magnitude of the vector projection 


Term
What are traces of a surface and how do you find them? 

Definition
The result of finding the surface on any plane
solving for any of the variables of the surface being constant 


Term
What is the standard form equation of ellipsoids? 

Definition
(x^{2}/a^{2}) + (y^{2}/b^{2}) + (z^{2}/c^{2}) = 1



Term
What is the standard form equation for a cone? 

Definition
(z^{2}/c^{2}) = (x^{2}/a^{2}) + (y^{2}/b^{2})



Term
What is the standard form equation for an elliptic parabaloid? 

Definition
(z/c) = (x^{2}/a^{2}) + (y^{2}/b^{2})



Term
What is the standard form equation for a hyperboloid of one sheet? 

Definition
(x^{2}/a^{2}) + (y^{2}/b^{2})  (z^{2}/c^{2}) = 1



Term
What is the standard form equation for a hyperbolic paraboloid? 

Definition
(z/c) = (x^{2}/c^{2})  (y^{2}/b^{2})


