Term
| What is the difference between a vector and a scalar? |
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Definition
vector = had magnitude and direction
scalar = has only magnitude |
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Term
| How do you add two vectors geometrically? |
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Definition
| by putting them end to end such that one starts where the other ends |
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Term
| How do you add two vectors algebraically? |
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Definition
| by finding the distance between the start point of the first and the end point of the second using the distance formula |
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Term
| If a is a vector and c is a scalar, how is ca related to a geometrically? |
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Definition
| ca has the same direction as a but has a magnitude that is c times greater or smaller than a |
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Term
| If a is a vector and c is a scalar, how do you find ca algebraically? |
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Definition
| multiply the components of vector a each by the scalar c |
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Term
| How do you find the vector from one point to another? |
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Definition
if the points are P1= (x0,y0,z0) and P2=(x1,y1,z1) then the vector v=< x1-x0 , y1-y0 ,z1-z0 >
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Term
| How do you find the dot product of a • b of two vectors if you know their lengths and the angle between them? |
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Definition
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Term
| How do you find the dot product of a • b of two vectors if you know their components? |
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Definition
if a = < a1, b1, c1> and b = <a2, b2, c2>
a • b = a1a2 + b1b2 + c1c2 |
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Term
| What is the expression for the scalar projection of b onto a? |
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Definition
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Term
What is the expression for the vector projection of b onto a?
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Definition
projab = (a • b)/|a|2*a
or
projab = compab * a/|a| |
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Term
| How do you find the cross product a x b of two vectors if you know their lengths and the angle between them? |
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Definition
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Term
| How do you find the cross product a x b of two vectors if you know their components? |
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Definition
if a = < a1, b1, c1> and b = <a2, b2, c2>
a x b = < b1c2 - c1b2 , c1a2 - a1c2 , b1a2 - a1b2 > |
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Term
| How do you find the area of the parallelogram determined by a and b? |
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Definition
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Term
| How do you find the volume of the parallelepiped determined by a, b, and c? |
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Definition
| V = scalar triple product = |a • (b x c)| |
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Term
| How do you find a vector perpendicular to a plane? |
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Definition
| find the normal vector, which is the cross product of two vectors lying on the plane |
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Term
| How do you find the angle between two intersecting planes? |
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Definition
| by finding the dot or cross product of the intersecting planes normal vectors and solving using the magnitude angle relationship |
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Term
| How do you write the vector equation for a line? |
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Definition
r = r0 + tv
where r0 is a point on the line and v is a vector parallel to the line and the line depends on the scalar t |
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Term
| How do you write parametric equations for a line? |
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Definition
v = < a, b, c>
r0 = < x0, y0, z0>
x = x0 + ta
y = y0 + tb
z = z0 + tc |
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Term
| How do you write symmetric equations for a line? |
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Definition
v = < a, b, c>
r0 = < x0, y0, z0>
(x - x0)/a = (y - y0)/b = (z - z0)/c
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Term
| How do you write vector equations for a plane? |
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Definition
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Term
| How do you write scalar equations for a plane? |
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Definition
n = < a, b, c >
r = < x, y, z >
r0 = < x0, y0, z0 >
a(x - x0) + b(y - y0) + c(z - z0) = 0
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Term
| How do you tell if two vectors are parallel? |
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Definition
| if the cross product is zero |
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Term
| How do you tell if two vectors are perpendicular? |
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Definition
| if the dot product is zero |
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Term
| How do you tell if two planes are parallel? |
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Definition
| if the cross product of their normal vectors is zero |
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Term
| How do you find the distance from a point to a line? |
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Definition
| find the magnitude of the projab if a is a vector on the line and b is a vector from a point on the line to the point |
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Term
| How do you find the distance from a point to a plane? |
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Definition
use the distance formula
plane: 0 = ax + by + cz + d
point: (x0, y0, z0)
D = |ax0 + by0 + cz0 + d|/√(a2 + b2 + c2) |
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Term
| How do you find the distance between two lines? |
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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 |
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Term
| What are traces of a surface and how do you find them? |
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Definition
The result of finding the surface on any plane
solving for any of the variables of the surface being constant |
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Term
| What is the standard form equation of ellipsoids? |
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Definition
(x2/a2) + (y2/b2) + (z2/c2) = 1
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Term
| What is the standard form equation for a cone? |
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Definition
(z2/c2) = (x2/a2) + (y2/b2)
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Term
| What is the standard form equation for an elliptic parabaloid? |
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Definition
(z/c) = (x2/a2) + (y2/b2)
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Term
| What is the standard form equation for a hyperboloid of one sheet? |
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Definition
(x2/a2) + (y2/b2) - (z2/c2) = 1
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Term
| What is the standard form equation for a hyperbolic paraboloid? |
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Definition
(z/c) = (x2/c2) - (y2/b2)
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