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Jessica Agbolade March 2021
• 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
An Intro to Vectors
tail
tip
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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
Vector Applications
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
cross product memory aid:
Find the volume of the parallelepiped whose edges are
a =2i-3j+4k
b =i+2j-k
c = 2i-j+2k
Equations of Lines and Planes
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
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]
The Relationship Between Points, Lines and Planes
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
Intersection of 2 Lines
1) Lines intersect at a point - one solution
2) Lines are coincident - infinite solutions
3) Lines are parallel - no solutions
1) intersect at one point
2) coincident
3) parallel
4) They are skewed lines - no solutions
Intersection of a Line and a Plane
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
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
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
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.