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Vector Calculus Study Guide: Vectors, Addition, Dot & Cross Product, Geometry - Prof. Char, Study notes of Analytical Geometry and Calculus

Louisiana State University (LSU) - System OfficeAnalytical Geometry and Calculus
Prof. Charles Richar Egedy

This vector calculus study guide by charlie egedy covers the fundamental concepts of vectors in r3, vector addition, scalar multiplication, dot product, cross product, and their geometric interpretations. It includes definitions, theorems, and examples.

Typology: Study notes

Pre 2010

Uploaded on 10/27/2009

jdanti4
jdanti4 🇺🇸

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Vector Calculus Study Guide
Charlie Egedy
April 7, 2009
1 Vectors in R3
Note that to get definitions and theorems for R2, simply make the third coordinate zero, since this will
restrict attention to the x-y coordinate plane in R3.
•If (a, b, c) is a point in R3, then ha, b, ciis a position vector, that is, one that begins at the origin
and ends at the point, with orientation toward the point.
•Given two points, we can define a vector whose coordinates are computed as the differences
between corresponding coordinates. Orientation is away from the point whose coordinates were
subtracted.
•We define vector addition by adding corresponding coordinate, and scalar multiplication by mul-
tiplying each coordinate by the same scalar. Thus, ha, b, ci+rhc, e, f i=ha+rd, b +r e, c +rf i.
•Vector addition has the geometric interpretation of constructing the diagonal of a parallelogram
defined by the two vectors, oriented outward from the common point of origin. The difference
between two vectors is the other diagonal, with orientation so that the convention on vector
addition is maintained.
•Vector addition is commutative and associative, there is a zero vector, and each vector has an
additive inverse. Additionally, scalar multiplication is distributive over vector addition.
•If ~a =ha1, a2, a3i, then k~ak=pa2
1+a2
2+a2
3.
•Given any vector other than the zero vector, we can define a unit vector having the same orien-
tation as the given vector. Thus, given ~v, the corresponding unit vector is e~v =~v
k~vk.
•Sometimes it will be useful to use the notation ˆ
i=h1,0,0i,ˆ
j=h0,1,0iand ˆ
k=h0,0,1i.
•Triangle inequality always applies: k~v +~wk≤k~vk+k~wk.
•We orient R3according to the right hand rule.
•An equation for a line can be written as a vector valued function of position vectors that are
oriented toward points on the line by writing ~r(t) = hx0, y0, z0i+thv1, v2, v3i. The first vector is a
position vector that hits some point on the line and the second vector gives the line its orientation.
•The parametric form of the equation for a line comes from writing the coordinates explicitly. Thus
x=x0+tv1,y=y0+tv2and z=z0+tv3.
•The symmetric form for a line comes by equating tin the above equations. Thus, x−x0
v1=y−y0
v2=
z−z0
v3. If any of the coordinates for the orienting vector is zero, this means that the cooresponding
coordinate does not change with t, so we leave that variable out of the string of equalities, writing
the constant that that coordinate is equal to apart from the equation.
•Given points P and Q, we can parameterize a line by writing ~r(t) = (1 −t)~
OP +t~
OQ. The point
midway between P and Q then occurs when t=1
2.
1
pf3
pf4

Partial preview of the text

Download Vector Calculus Study Guide: Vectors, Addition, Dot & Cross Product, Geometry - Prof. Char and more Study notes Analytical Geometry and Calculus in PDF only on Docsity!

Vector Calculus Study Guide

Charlie Egedy

April 7, 2009

1 Vectors in R

Note that to get definitions and theorems for R^2 , simply make the third coordinate zero, since this will restrict attention to the x-y coordinate plane in R^3.

  • If (a, b, c) is a point in R^3 , then 〈a, b, c〉 is a position vector, that is, one that begins at the origin and ends at the point, with orientation toward the point.
  • Given two points, we can define a vector whose coordinates are computed as the differences between corresponding coordinates. Orientation is away from the point whose coordinates were subtracted.
  • We define vector addition by adding corresponding coordinate, and scalar multiplication by mul- tiplying each coordinate by the same scalar. Thus, 〈a, b, c〉 + r 〈c, e, f 〉 = 〈a + rd, b + re, c + rf 〉.
  • Vector addition has the geometric interpretation of constructing the diagonal of a parallelogram defined by the two vectors, oriented outward from the common point of origin. The difference between two vectors is the other diagonal, with orientation so that the convention on vector addition is maintained.
  • Vector addition is commutative and associative, there is a zero vector, and each vector has an additive inverse. Additionally, scalar multiplication is distributive over vector addition.
  • If ~a = 〈a 1 , a 2 , a 3 〉, then ‖~a‖ =

a^21 + a^22 + a^23.

  • Given any vector other than the zero vector, we can define a unit vector having the same orien- tation as the given vector. Thus, given ~v, the corresponding unit vector is e~v = (^) ‖~~vv‖.
  • Sometimes it will be useful to use the notation ˆi = 〈 1 , 0 , 0 〉, ˆj = 〈 0 , 1 , 0 〉 and kˆ = 〈 0 , 0 , 1 〉.
  • Triangle inequality always applies: ‖~v + w~‖ ≤ ‖~v‖ + ‖ w~‖.
  • We orient R^3 according to the right hand rule.
  • An equation for a line can be written as a vector valued function of position vectors that are oriented toward points on the line by writing ~r(t) = 〈x 0 , y 0 , z 0 〉 + t 〈v 1 , v 2 , v 3 〉. The first vector is a position vector that hits some point on the line and the second vector gives the line its orientation.
  • The parametric form of the equation for a line comes from writing the coordinates explicitly. Thus x = x 0 + tv 1 , y = y 0 + tv 2 and z = z 0 + tv 3.
  • The symmetric form for a line comes by equating t in the above equations. Thus, x− v 1 x 0 = y− v 2 y 0 = z−z 0 v 3. If any of the coordinates for the orienting vector is zero, this means that the cooresponding coordinate does not change with t, so we leave that variable out of the string of equalities, writing the constant that that coordinate is equal to apart from the equation.
  • Given points P and Q, we can parameterize a line by writing ~r(t) = (1 − t) OP~ + t OQ~. The point midway between P and Q then occurs when t = 12.
  • We can write the equation of a sphere having radius R, centered at (a, b, c) by (x − a)^2 + (y − b)^2 + (z − c)^2 = R^2.
  • We define dot product between two vectors as the sum of corresponding products. Thus, ~v · w~ = v 1 w 1 + v 2 w 2 + v 3 w 3.
  • The dot product of a vector with itself is the square of its magnitude.
  • The angle between vectors is given by cos θ = (^) ‖~v~v‖‖·^ w~ w~‖. If the numerator is positive, the angle is acute; if negative the angle is obtuse. If zero, the angle is right. In the last case we say that vectors are perpendicular or orthogonal.
  • We can resolve one vector in terms of another by first computing the magnitude of the parallel resolution and then multiplying this by a unit vector in the desired direction. Thus comp (^) w~(~v) = ~v· w~ ‖ w~‖ is the component, and the projection itself is proj^ w~(~v) =^

~v· w~ ‖ w~‖ e^ w~, as the parallel component of ~v in the direction of w~. We compute the perpendicular component of the resolution by subtracting the parallel component from the original vector being resolved.

  • The cross product of two vectors is a vector that is perpendicular to the two given vectors. We

define the cross product as the determinant of a matrix. Thus, ~u × ~v =

ˆi ˆj kˆ u 1 u 2 u 3 v 1 v 2 v 3

  • The magnitude of the cross product is the product of the magnitudes of the two vectors times the sine of the angle between them. The orientation of the cross product is given by the right hand rule.
  • Cross product is anticommutative (that is, swapping the order induces a minus sign). The cross product of a vector with itself is the zero vector, and cross product distributes over addition.
  • The area of a parallelogram is the magnitude of the cross product of the two vectors that define it. The volume of a parallelpiped is the triple product of the three vectors that defines it.
  • The equation ax + by + cz = d is a plane with normal 〈a, b, c〉 containing the point (x 0 , y 0 , z 0 ), where ax 0 + by 0 + cz 0 = d.
  • Angles between planes are defined to be the angles between their normals.
  • Parallel planes have parallel normals.
  • To find the point of intersection between a line and a plane, plug the parametric equations of the line into the line into the equation of the plane and solve for t.
  • To find the point of intersection of two lines, give them different parameter names and then equate the corresponding coordinates, solving for the two parameters. If no solution exists, then the lines are parallel (with parallel orienting vectors) or they are skew.
  • The distance between two planes is the projection of a vector connecting points in each plane with a common normal for the two planes.
  • The distance between skew lines is the projection of a vector connecting points in each line with a normal defined to be the cross product of the vectors orienting the two lines.
  • To find the equation of a line of intersection between two planes, first compute the orienting vector as the cross product of the normals for the two planes. Then find a point common to the two planes by solving the equations of the plane simultaneously, choosing any point of convenience.
  • Given three points, the equation of the plane containing them can be found by first defining two vectors oriented in the plane as differences between point coordinates, computing the cross product of these two vectors to get the plane’s normal, then computing d using one of the three points.

3 Partial Derivatives

  • A partial derivative is a derivative of a function having more than one independent variable, taken by treating all but one of the variables as constants and then taking the derivative with respect to the nonconstant variable.
  • If derivatives are continuous, then mixed partials can be computed by taking derivatives in any order.