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Vectors

~ a visual representation of the course

Jessica Agbolade March 2021

In This Mind Map I will...

Vectors

• Distinguish between the geometric representations of a single linear equation or a

system of two linear equations in two-space and three-space, and determine different

geometric configurations of lines and planes in three-space;

• Represent lines and planes using scalar, vector, and parametric equations and solve

problems involving distances and intersections

• Demonstrate an understanding of vectors in two-space and three-space by representing

them algebraically and geometrically and by recognizing their applications

• Perform operations on vectors in two-space and three-space, and use the properties of

these operations to solve problems, including those arising from real-world applications

Chapter 6

An Intro to Vectors

Chapter 6

Key Terms

  • Scalar (k): a quantity with magnitude
  • Vector (AB): a quantity with magnitude and direction
  • Equal Vectors: same magnitude and direction
  • Opposite Vectors: same magnitude, opp. direction (aka negative of a vector)
  • Zero Vector: results from adding two opposite vectors
  • Unit Vector: a vector with magnitude 1. Any vector can be represented by the sum of its unit vectors
  • i = unit vector for positive x axis
  • j = “ positive y axis
  • k = “ positive z axis
  • Collinear Vectors: parallel to the same line
  • Position Vector (OP): starts at the origin (O) and ends at a point (P)
  • Equilibrant Vector (e): opposite the resultant vector, net force 0
  • Linear Combination of Vectors: formed by adding scalar multiples of 2 or more vectors.

Formulas

Concepts

tail

tip

Concepts

r

  • Vectors are drawn as arrows with a tip and tail

  • To add 2 vectors, position them tip to tail. This forms an open triangle. completing the triangle gives you the sum of the two vectors aka the resultant vector (r)
  • Subtract vectors by adding the opposite vector
  • Vectors follow commutative, associative, and identity properties
  • They can be added in any order
  • Simplifying vector expressions is similar to simplifying integer expressions

  • When a vector is multiplied by a scalar, the magnitude is multiplied by the scalar and the vectors are parallel.
  • Directions remain unchanged if the scalar is positive, and becomes opposite if scalar is negative.
  • Multiplying vectors follow the distributive and associative rule
  • They can be expanded using FOIL
  • They can be multiplied in any order

The vectors d and e are such that |d| = 3 and |e| = 5, and the angle between them is 30°. Determine:

a) |d + e|

b) |d − e|

c) a unit vector in the direction of d + e

Example

Chapter 7

Vector Applications

Chapter 7

Key Terms

Force (F): any interaction that, when unopposed, will change the motion of an object. Force is a vector quantity measured in kg*m/s^2 aka Newtons (N)

Resultant Force: single force used to represent the components of multiple forces acting on an object

Equilibrant Force: opposite the resultant force

Resolution of a vector into its components: split apart a force into its components

Air/Water Speed: speed of a plane/boat in the air/water

Ground Speed: speed of a plane from the POV of someone watching from the ground

Dot Product: multiplication of vectors by vectors

Direction Angles: angles that OP makes with the x,y and z axis

Cross Product: gives a vector that is perpendicular to the original vectors (only in R3)

Work (W): how much energy is put on an object for it to move. Work is a scalar quantity measured in N*m = Joules (J)

Torque (T): a rotating force measured in Joules. Torque is a work applied in a perpendicular direction to the original forces

Formulas

Concepts

  • Sometimes it's useful to split apart a force into its x and y components (resolution of a vector Into its components). Best way to do that is by having the components make an angle of 90 with each other

  • For any 2 vectors u and v with angle theta between them, the projection of v on u is the vector component of v in the direction of u.
  • The vector projection of one vector onto another is calculated by multiplying the scalar projection by the unit vector
  • Based on the angle between u and v, you can determine what direction a projection will be

Concepts (2)

cross product memory aid:

Cont'd

  • Right hand rule to find direction: Fingers point towards vector r, then bend towards vector f, thumb points in direction of the cross product
  • The cross product must be done before the dot product is done

  • Cross product also calculates the area of a parallelogram
  • Torque is the cross product between the length of the wrench and the force applied

Find the volume of the parallelepiped whose edges are

a =2i-3j+4k

b =i+2j-k

c = 2i-j+2k

Example

Chapter 8

Equations of Lines and Planes

Chapter 8

Key Terms

Direction Vector (m): a non zero vector parallel to the given line. It is the slope of the line in vector form

Normal Vector (n): line perpendicular to the given line

Coincident Lines: lines that lie on top of each other, equivalent

Plane (pi): a flat surface that extends infinitely in all directions. Represented by a parallelogram

EOL: Equation Of a Line

EOP: Equation Of a Plane

Formulas

Concepts

  • In R2 a line can be defined in 4 ways: Slope y-intercept form, vector form, parametric form, or cartesian/scalar form
  • To determine the EOL we need 2 points or the slope and a point

  • 2 lines are parallel iff they are scalar multiples of each other (n and m would also be parallel and scalar multiples)
  • 2 lines are perpendicular iff their dot product equates to 0 (n dot m=0)

  • In R3 a line can be defined by a vector, parametric or symmetric equation

  • To determine the EOL we need one of the following
  • A point and 2 direction vectors
  • 3 points
  • A point and a line or
  • 2 intersecting/ parallel lines

Concepts (2)

Cont'd

  • The angle between planes is equal to the angle between n
  • cosX= (n1 dot n2)/(|n1||n2|)
  • 2 planes are parallel when their normals are parallel
  • n1=kn2
  • 2 planes are perpendicular when their normals are perpendicular
  • n1 dot n2=0

1) Determine a vector equation for each line.

a) perpendicular to line 4x − 3y = 17 and through point P(−2,4)

b) parallel to the z-axis and through point P(1,5,10)

c) parallel to [x, y, z] = [3,3,0] + t[3,−5,−9] with x-intercept of −10

d) with the same x-intercept as [x, y, z] = [3,0,0] + t[4, −4,1] and the same z-intercept as [x, y, z] = [6,−2, −3] + t[3,−1, −2]

2) Determine if the point P(−4,−13,10) is on the plane [x, y, z] = [6,−7,10] + s[1,3,−1] + t[2,−2,1]

Example

Chapter 9

The Relationship Between Points, Lines and Planes

Chapter 9

Key Terms

Skew Lines: two lines that do not intersect and are not parallel, exist in R3

Coplanar: points/ lines that lie in the same plane

Consistent System: a system with at least one solution

Inconsistent System: a system with no solutions

Formulas

Concepts

Intersection of 2 Lines

  • In R2, the intersection of 2 lines has 3 possibilities:

1) Lines intersect at a point - one solution

2) Lines are coincident - infinite solutions

3) Lines are parallel - no solutions

Concepts

  • In R3, the intersection of 2 lines has 4 possibilities

1) intersect at one point

2) coincident

3) parallel

4) They are skewed lines - no solutions

Concepts (2)

Intersection of a Line and a Plane

  • In 3-Space, the intersection of planes and lines have 3 possibilities

1) The line intersects the plane at a point - one solution

2) The line is coincident with the plane - infinite solutions

3) The line is parallel and distinct from the plane - no solutions

LINE + PLANE

Concepts (3)

Intersection of 2 Planes

  • There are 3 possibilities for the intersection of 2 planes

1) The planes intersect at a line - one line of intersection

2) The planes are coincident - infinite solutions

3) The planes are parallel - no solutions

PLANE + PLANE

Concepts (4)

Intersection of 3 Planes

  • There are 7 possibilities for intersection of 3 planes:

If their normals are parallel

1) All 3 planes are coincident if normals are all the same

2) All 3 planes are parallel if normals are all distinct

3) 2 Planes are coincident and 1 is parallel if 2 normals are distinct

4) 1 Plane intersects 2 planes that are coincident/parallel if 2 normals are the same

3 PLANES

If their normals are not parallel

If triple scalar product does not equal 0

5) A Triangular prism is formed and no intersection (0x + 0y = c)

6) A “revolving door” is formed with a line intersection (0x + 0y = 0)

If triple scalar product = 0

7) All planes intersect at a point

Determine if the lines intersect:

L1: [x,y,z] = [7,2,-6] + s[2,1,-3]

L2: [x,y,z] = [3,9,13] + t[1,5,5]

If they do, find the coordinates of the point of intersection.

Example

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