VECTORS

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Equations of lines and planes

Sketching Planes

Using a line and multiple points we can come up with the equation of a plane

Vector and parametric equations of a plane

These equations can allow us to obtain any point on the plane

Can be determined in 4 ways

Cartesian equations of planes

need a point and a vector to derive the equation

Written in the form: Ax + By + Cz + D = 0

Cartesian equations of lines

Written in the form: Ax + By + C = 0

Symmetric Equations of lines in R3

Derived from parametric equations

Vector and parametric equations of lines

To find equations of lines we must be given two distinct points/one point along with a direction vector

Applications of Vectors

z

Finding torque

dot product is used when installing solar panels

Calculating work done

Scalar and Vector projects

A real life application includes computer animation.

Direction Angles

Projection of one vector onto another. You can think of it kind of like a shadow

Cross Product

Multiple methods are available for you to find the cross product. Find which one works best for you!

Finds the orthogonal of the vectors

Dot Product

Tells us the amount of force applied in the direction of motion

Vectors should be placed tail to tail

Angle x must be from 0 to 180 degrees

Forces and Vectors

Equilibrant of multiple forces

resultant and composition of forces yields combined force

Vectors can be expressed as forces and velocity

VECTORS

Learn more about how a small village on the banks of the Tiber River became the core of one of the most powerful ancient civilizations.

Intro to Vectors

Linear combinations of vectors

vectors in the form a(scalar)i(vector + b(scalar)j(vector) + c(scalar)k(vector) in R2

vectors in the form a(scalar)i(vector + b(scalar)j(vector) in R2

Operations of vectors in R3

Finding position vectors in R3

Addition of vectors using a method such as the parallelogram law

Using Pythagorean theorem to define two points

Operations of vectors in R2

Scalar multiplication using components

Addition using component form

Vectors in R2 and R3

Different planes in R3 such as the XY plane or the YZ plane or the XZ

Points of vectors (X, Y) / (X, Y, Z)

Position Vectors

Properties of vectors

distributive property of addition

Associative property of addition

Commutative property

Multiplication of vectors and scalars

Adding and subtracting Vectors

Difference between vectors

Parallelogram law

Triangle law of addition

Zero Vectors

Terms

Opposite Vectors

Equal Vectors

Vectors

Scalars

Geometric Vectors

Additional real world Applications of Vectors

sources: 1. https://testbook.com/learn/maths-application-of-vector/ 2. https://www.slideshare.net/Iftekharbhuiyan1/real-life-application-of-vector

Roller coasters

Navigating in air and water often uses vectors

The swinging of a pendulum

Sharpening pencil with a blade