MIchael Driesen
Mrs. Rozell
Math 10H
17 December 2011
Vectors
Math is everywhere. No matter which way you look at it, it’s there. It is especially present in science. Most people don’t notice it, they have to look closer to find out what it is really made of. A component in math that is very prominent in science is the vector. What is a vector? A vector is a geometric object that has both a magnitude and a direction. A good example of a vector is wind. 30 MPH north. It has both magnitude,(in this case speed) and direction. Vectors have specific properties that make them very useful in real life applications. Through the use of these special objects, many advancements in the fields of math and science are available. Representations

Vectors can sometimes be hidden behind basic objects. They are usually represented with an arrow on top of its starting point and terminal point, as shown here: The most common form of vector is the bound vector. All that means is that the starting point of the vector is the origin, or (0,0). The bound vector goes from the origin to it’s terminal point, which in this case can be (3,4). An easy way to write this is A = (3,4), where A is the vector. On a graph, it looks like this:

That applies for two-dimensional vectors. Three dimensional vectors can be represented that way also. In a 3 dimensional vector, its terminal point has coordinates in three axis. These points can be represented by the letters i, j, and k. The letter i is for the x axis, j for the y axis, and k for the z axis. An example of a 3 dimensional vector is written as: A = 6i+3j+4k. It can also be denoted as A = (6,3,4). Where (6,3,4) is the terminal point. Two vectors are equal if their coordinates are equal, so if another vector, say B, had the terminal point (6,3,4), and if both vectors are bound vectors, they would be equal. Aside from direction, a vector has one other part, which is magnitude. Magnitude is denoted by the original vector...

...ECE 352 VECTOR ANALYSIS
DEL OPERATOR
GROUP 3 Andaya, Rizalyn Ramos, Maria Issa P.
∇
Del is a symbol used in mathematics, in particular, in vector calculus, as a vector differential operator, usually represented by the nabla symbol ∇. Del may denote the gradient (locally steepest slope), the divergence of a vector field, or the curl (rotation) of a vector field. The symbol ∇ can be interpreted as avector of partial derivative operators, and its three possible meanings—gradient, divergence, and curl—can be formally viewed as the product of scalars, dot product, and cross product, respectively, of the del "operator" with the field.
A. GRADIENT If a scalar function , , is continuously differentiable with respect to its variables x, y, z through the region, then the gradient of , written grad , is defined as the vector grad = Using the vector differential operator ∇, ∇= We can write the gradient of as
+
+
+
+
grad = ∇ = ( = Example: If =
+ +
+ +
)
, determine the grad at the point P(1,3,2).
By definition: grad = ∇ =
+
+
All we have to do then is to find the partial derivatives at x=1, y=3, z=2 and insert their values.
Since = = =
= = = = 2(1) (3) (2)3 + (3)2(2)2 (1)2(2)3 + 2(1) (3) (2)2 3(1) (3) (2)2 + 2(1) (3)2(2) = 84 = 32 = 72
grad
= ∇ = 84 i + 32 j +72 k
B. DIVERGENCE If a vector...

...HL Vectors Notes
1.
Vector or Scalar
Many physical quantities such as area, length, mass and temperature are completely described once the magnitude of the quantity is given. Such quantities are called “scalars.” Other quantities possess the properties of magnitude and direction. A quantity of this kind is called a “vector” quantity. Winds are usually described by giving their speed and direction; say 20 km/h north east. The wind speed and wind direction together form a vector quantity called the wind velocity. A force, for example, is characterized by its magnitude and direction of action. The force would not be completely specified by one of these properties without the other. The velocity of a moving body is determined by its speed and direction of motion. Acceleration and displacement are other examples of vector quantities.
A scalar is a quantity that has magnitude or size but no direction.
A vector is a quantity that has both magnitude and direction.
Displacement Vectors and Notation
[pic]
Vectors can be represented geometrically by arrows in 2- or 3-space; the direction of the arrow specifies the direction of the vector and the length of the arrow describes its magnitude. The first point in the arrow is called the initial point of the vector and the tip is called the terminal point. We shall denote...

...3r21. ABCD is a rectangle and O is the midpoint of [AB].
Express each of the following vectors in terms of and
(a)
(b)
(c)
(Total 4 marks)
2. The vectors , are unit vectors along the x-axis and y-axis respectively.
The vectors = – + and = 3 + 5 are given.
(a) Find + 2 in terms of and .
A vector has the same direction as + 2 , and has a magnitude of 26.
(b) Find in terms of and .
(Total 4 marks)
3. The circle shown has centre O and radius 6. is the vector , is the vector and is the vector .
(a) Verify that A, B and C lie on the circle.
(3)
(b) Find the vector .
(2)
(c) Using an appropriate scalar product, or otherwise, find the cosine of angle .
(3)
(d) Find the area of triangle ABC, giving your answer in the form a , where a ∈ .
(4)
(Total 12 marks)
4. The quadrilateral OABC has vertices with coordinates O(0, 0), A(5, 1), B(10, 5) and C(2, 7).
(a) Find the vectors and .
(b) Find the angle between the diagonals of the quadrilateral OABC.
(Total 4 marks)
5. Find a vector equation of the line passing through (–1, 4) and (3, –1). Give your answer in the form r = p + td, where t ∈ R
(Total 4 marks)
6. In this question, the vector km represents a displacement due east, and the vector km a...

...Chapter 1 Vectors, Forces, and Equilibrium
1.1 Purpose
The purpose of this experiment is to give you a qualitative and quantitative feel for vectors and forces in equilibrium.
1.2
Introduction
An object that is not accelerating falls into one of three categories: • The object is static and is subjected to a number of diﬀerent forces which cancel each other out. • The object is static and is not being subjected to any forces. (This is unlikely since all objects are subject to the force of gravity of other objects.) • The object is moving with constant velocity. In this case, the object may be subject to a number of forces which cancel out or no force at all. This case is not considered in this lab. The category of physics problems that involve forces in static equilibrium is called statics. Physicists and engineers are subjected to static problems quite frequently. A few examples of these principles in use are seen in the design of bridges and the terminal velocity of a person falling through the air. Mathematically, forces in equilibrium are just a special case of Newton’s Second Law of Motion, which states that the sum of all forces is equal to the mass of the object multiplied by the acceleration of the object. The special case of forces in equilibrium (static), occurs when the acceleration of the object is zero. When this situation arises, Newton’s Law becomes: ΣF = 0 (1.1)
This equation simply states that the sum of all of...

... Date 09/14
PHET 2. Vector Addition
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Learn how to add vectors. Drag vectors onto a graph, change their length and angle, and sum them together. The magnitude, angle, and components of each vector can be displayed in several formats.
Learning Goals
• Explain vector representations in their own words.
• Learn about the polar form of vectors and the component form.
Introduction:
A vector quantity is one that has both a magnitude and a direction. For instance, a velocity vector will have a magnitude (24 m/s) and a direction (northeast or 45 degrees). These simulations will demonstrate how vectors can be summed to produce a resulting vector, and how the acceleration vector affects the velocity vector.
[pic]
Part I: Vector Addition Simulation: [pic]
Place two vectors [pic]in the work area. Change their direction and magnitude be dragging the heads of the arrows representing each vector. Click [pic] to view the resultant (sum) of the two vectors. You may click the Styles to show the X and Y components.
Click on one vector and fill in the boxes: [pic]
Click on another...

...Research paper about vector organisms such as ticks and mosquitoes and ways to remove them.
Vector Organisms and the Diseases They Carry
Vector organisms are parasites that transmit various diseases. Two vector organisms are ticks and mosquitoes. These two vector organisms transmit many well-known diseases. Through most of this essay, I will be explaining these two different organisms and the diseases that they carry.
First, I will be explaining deer ticks. Ticks are small spider-like organisms that attach themselves to a donor body. Male ticks are black. Females have a red abdomen and some black near the head. Adult ticks usually feed on deer but some times feed on humans accidentally. Deer ticks go through three life stages. The first stage is the larva. The larvae are small and tan colored. They usually feed on things like chipmunks, mice, and other small animals. Nymphs are a little bit larger than the larva. They are beige with a dark head. Nymphs usually feed on larger animals such as birds and raccoons. The adults vary in color depending if they are males or females. They usually feed on cats, dogs, and humans. Deer ticks are usually found in grassy woodland areas or wide open plains. Deer ticks are more common in the eastern U.S. than in the western U.S.
Mosquitoes are small insects that have a very thin body. Mosquitoes can be found anywhere except for Antarctica. There are about 170...

...
Abstract
The experiment was about the resolution of vector quantities using different methods or techniques. Among those are the Parallelogram, Polygon, and the Analytical or Component methods. Using each method, it was found out that Component method is the most accurate as its approach is purely theoretical, that is, all other physical factors are neglected leaving only the appropriate ones to be calculated. In addition, properties of these quantities such as associativity and commutativity of the addition operation were also explored.
1. Introduction
Vectors play an important role in many aspects of our everyday lives or of one’s daily routine. It is a mathematical quantity that has both a magnitude and direction.
A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier". The study of vectors had gone through a lot of revisions, starting from the 19th century where mathematicians used geometrical representations for complex numbers. Lots of changes and multiple varieties of altering were conducted to this study, which led to the discovery of the vector that we all know today. Operations on vectors are also made possible through time. Addition of vectors was clarified and can now be done in different ways. Vector addition in a graphical way can use the polygon method and the...