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Vectors - Coggle Diagram
Vectors
Chapter 6: Intro to Vectors
Intro to Vectors
Geometric Vectors
the tail is a→, the end is b→. |ab| represents the magnitude of vector ab→
When drawing a vector, it will have a tail, A→, (initial point) and a head, B→, (final point). The magnitude of a vector is described by its length.
Algebraic Vectors
Vectors that are linked to a point in a coordinate system and whose components are derived from the reference system (i.e. 2D or 3D cartesian plane).
Equal Vectors
If two vectors have the same magnitude (size) and the same direction, then we call them equal to each other.
For example, if we have two forces, 𝐹1→=30 N in the upward direction and 𝐹2→=30 N in the upward direction, then we can say that 𝐹1→=𝐹2→.
Opposite Vectors
In the same way that scalars may have positive or negative values, vectors can as well. A negative vector is a vector that points in the opposite direction of the positive reference vector.
For example, if in a particular situation, we define the upward direction as the reference positive direction, then a force 𝐹1→=30 N downwards would be a negative vector and could also be written as 𝐹1→=−30 N. In this case, the negative sign (−) indicates that the direction of 𝐹1→ is opposite to that of the reference positive direction.
Vector Addition/Multiplication
Parallelogram Law: Vector Addition a + b.
The resulting vector a + b is formed by adding the vectors a and b from tip to tail. This resultant forms the diagonal of a parallelogram that can be formed by the addition
Difference of Two Vectors a − b
Arrange vectors tail-tail
Subtraction may be rewritten as a + (-b), and the vectors can be added tip-tail , with vector -b pointing in the opposite direction of b.
Multiplication of a Vector by a Scalar (k)
Vectors may be multiplied in the same way that they can be added and subtracted to modify their magnitude and direction.
When a vector an is multiplied by a scalar k>0, ka moves in the same direction as a.
If we multiply a vector a by a scalar k<0, ka will move in the opposite direction as an if 0 < k < 1, the magnitude of the vector a will be reduced; if k>1, the magnitude will be greater than a.
Collinear Vectors
Two vectors u and v are collinear if and only if a non-zero scalar, k, can be found such that u = kv, meaning that the two vectors are scalar multiples of each other.
Unit Vectors
The magnitude of the vector is 1: By dividing the magnitude of the vector by its components, we may get the unit vector of any vector.
Properties of Vectors
Commutative Property of Addition
Associative Property of Addition
Distributive Property of Addition
Associative Law for Scalars
Distributive Law for Scalars:
Vectors in R2 & R3
R2
The set of all 2 dimensional vectors is denoted R2. Algebraically, a vector in 3 (real) dimensions is defined to ba an ordered triple (x, y, z), where x, y and z are all real numbers (x, y, z ∈ R).
Operations
R3
The set of all ordered triples of real numbers is called 3‐space,.
Operations
Linear Combinations and Spanning Sets
Linear Combination of Vectors
Spanning Sets
If the vectors u and v are not co-linear, then the combination of these vectors spans R2, indicating that it is a Spanning Set.
A linear combination of these non-collinear vectors, u and v, is au + bv, where a and b are scalars. The diagonal of the parallelogram produced by the vectors au and bv is the vector au + bv.
Unit 7: Application of Vectors
Vectors as Forces & Velocity
Forces
f=ma
Mass (kg) and acceleration (m/s2) are multiplied to calculate force. The outcome is a Newtonian unit (N).
Resolution of a Vector;
f=(fx,fy)
Equilibrant Forces
When operating on an item, a number of vector forces resist each other. This force keeps the item in a condition of balance.
Velocity
Air Velocity: As an object travels, the velocity of air flow will create resultant velocity and/or change the direction of he object.
Ground Velocity: As an object travels, the velocity of water flow will create resultant velocity and/or change the direction of the object
The Dot Product of
Geometric/Algebraic Vectors
Scalar and Vector
Projections
The Cross Product
Applications of the Cross and Dot Product
Unit 8: Equations of Lines and Planes
Vector and Parametric Equations of a Line in R2
Vector Equation of a Line in R2: similar to equations of a line, vector equation requires a directional vector m = (a, b), and a vector ro⃗ from the origin (0,0) to a specific point Po(xo, yo).
vector equation: r=ro+tm;tER
Parametric: Allows to compartmentalize the x and y values of the vector equation.
x = x0 + ta
y = y0 − tb
Cartesian Equation of a Line
Equation of a line in standard form: Ax+By+c=0 where n=(A,B)
The Cartesian Equation of a Plane
The Cartesian Equation of a Plane follows the same principles as the Cartesian Equation of a Line;
Ax + By + Cz + D = 0, where n= (A, B,C), is a normal vector to this plane
:
Vector, Parametric, and Symmetric Equations of a Line in R3/Plane
Vector Equation: r=r0+tm;tER
x=x0+ta
y=yo+tb
z=zo+tc
Symetrical Equations of a line in R3:
x-xo/a=y-yo/b=z-zo/c : a,b,c =/ o
