# Shared Flashcard Set

## Details

Vector Concepts
Concept Check from Chapter 12 Early Transcendentals 6E
34
Mathematics
05/01/2011

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 P1= (x0,y0,z0) and P2=(x1,y1,z1) then the vector v=< x1-x0 , y1-y0 ,z1-z0 >
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
 a • b = |a|*|b|*cosΘ
Term
 How do you find the dot product of a • b of two vectors if you know their components?
Definition
 if a = < a1, b1, c1> and b =   a • b = a1a2 + b1b2 + c1c2
Term
 What is the expression for the scalar projection of b onto a?
Definition
 compab = (a • b)/|a|
Term
 What is the expression for the vector projection of b onto a?
Definition
 projab = (a • b)/|a|2*a or projab = compab * 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
 |a x b| = |a|*|b|*sinΘ
Term
 How do you find the cross product a x b of two vectors if you know their components?
Definition
 if a = < a1, b1, c1> and b = a x b = < b1c2 - c1b2 , c1a2 - a1c2 , b1a2 - a1b2 >
Term
 How do you find the area of the parallelogram determined by a and b?
Definition
 area = |a x b|
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 = 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
Term
 How do you write parametric equations for a line?
Definition
 v = < a, b, c> r0 = < x0, y0, z0> x = x0 + ta y = y0 + tb z = z0 + tc
Term
 How do you write symmetric equations for a line?
Definition
 v = < a, b, c> r0 = < x0, y0, z0> (x - x0)/a = (y - y0)/b = (z - z0)/c
Term
 How do you write vector equations for a plane?
Definition
 n • (r - r0) = 0
Term
 How do you write scalar equations for a plane?
Definition
 n = < a, b, c > r = < x, y, z > r0 = < x0, y0, z0 > a(x - x0) + b(y - y0) + c(z - z0) = 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 projab 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: (x0, y0, z0) D = |ax0 + by0 + cz0 + d|/√(a2 + b2 + c2)
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
 (x2/a2) + (y2/b2) + (z2/c2) = 1
Term
 What is the standard form equation for a cone?
Definition
 (z2/c2) = (x2/a2) + (y2/b2)
Term
 What is the standard form equation for an elliptic parabaloid?
Definition
 (z/c) = (x2/a2) + (y2/b2)
Term
 What is the standard form equation for a hyperboloid of one sheet?
Definition
 (x2/a2) + (y2/b2) - (z2/c2) = 1
Term
 What is the standard form equation for a hyperbolic paraboloid?
Definition
 (z/c) = (x2/c2) - (y2/b2)
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