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/Topic 1 Vectors & Matrix - Coggle Diagram
/Topic 1 Vectors & Matrix
Introduction to Vectors
Geometric Vectors
A(initial point) -> B(terminal point)
v = AB
Norm of A Vector
||v|| = √v1^2+v2^2 (2-space)
||v|| = √v1^2+v2^2+v3^2 (3-space)
Unit Vector
Unit Vector in Direction of v
u = v / ||v|| or (1 / ||v||)v
Standard Unit Vectors
v = <v1 , v2 , v3>
Dot Product
Component Form
u.v = u1v1 + u2v2 + u3v3
If θ is the angle between u and v can also used: u.v = ||u|| ||v|| cosθ
Properties
if u.v = o, then the two vectors are orthogonal/perpendicular to each other
Cross Product
Can be Calculates using:
u x v = <u2v3-u3v2, u3v1-u1v3, u1v2-u2v1> derived from the determinant notation of the two vectors
u x v does NOT equal to v x u
Properties
If u x v = 0, then the two vectors are parallel to each other
Planes & Lines in 3-Space
Planes
Point-Normal Equation of a Plane
a(x-x0) + b(y-y0) + c(z-z0)
Lines
a(x-x0) + b(y-y0) = 0
Parametric Equations
x = x0+at
y = y0+bt
z = z0+ct
Symmetric Equations
(x-x0)/a = (y-y0)/b = (z-z0)/c
Summary
How to Find Any Type of Equation (either Plane or Line)
Need to have 1 point and 1 vector
Example
Question: Find equation for the plane using points A, B, and C
Steps:
Find the vector of AB and AC
Find the cross product of the 2 vectors to get one vector <a,b,c>
Use point A (x,y,z) because it is common point use for finding vector
Substitute the point and the vector in the equation, a(x-x0) + b(y-y0) + c(z-z0)
A
Matrix
is a rectangular array of numbers, symbols, or expressions arranged in rows and columns.
The rows run horizontally, and the columns run vertically.
Each number, symbol, or expression in the array is called an element.
Addition and Subtraction
Matrix addition and subtraction are operations performed between two matrices of the same size. The addition or subtraction is done element-wise, meaning that each corresponding element in the matrices is added or subtracted.
Multiplication Matrix
Matrix multiplication is not commutative, meaning that A×BA×B is not necessarily equal to B×AB×A. Additionally, the number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be defined.