Vectors

Written by Elizabeth Robelo

Improved by Aparajita Satapathy

Improved by Lichao Tang

Improved by Jimin Yoon

Improved by Sabrina Seibel

In physics, a vector is a quantity with a magnitude and a direction.

The Main Idea

A vector is a quantity with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. In physics, a vector always starts at the source and directs to the observation location. Refer to the image below for a visual representation:

Visual Representation

We can also add and subtract vectors. To add two vectors you align them head to tail and make sure that the tails of both vectors coincide with each other. The new added vector is the connecting arrow starting from the tail of one to the head of the other vector.

Adding vector A to B

To subtract two vectors reverse the direction of the vector you want to subtract and continue to add them like shown before.

Subtracting vector B from A

A vector can also be multiplied by a scalar. To multiply a vector by a scalar we can strech, compress or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compresseed. If the scalar is greater than 1, the vector will get streched. If the scalar is a negative number, then the vector reverses its direction.

Scalar Multiplication

A Mathematical Model

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or (xi + yj - zk). Magnitude: [math]\displaystyle{ |A| = \sqrt{x^2 + y^2 + z^2} }[/math]

Dot Product: Ia·bI = IaI*IbI*cos([math]\displaystyle{ theta }[/math])

[math]\displaystyle{ \mathbf{A}\cdot\mathbf{B}=\sum_{i=1}^n A_iB_i=A_1B_1+A_2B_2+\cdots+A_nB_n }[/math]

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :[math]\displaystyle{ \mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|} }[/math]

Cross Product: Ia x bI =IaI*IbI*sin([math]\displaystyle{ theta }[/math])

a x b= <a1, a2, a3> x <b1, b2, b3> = <a2*b3 - a3*b2> i - <a1*b3 - a3*b1> j + <a1*b2 - a2*b1> k

Multiplication between vectors and scalars: c<a1, a2, a3> = <c*a1, c*a2, c*a3>

A Computational Model

In Physics 1, we have seen vectors in a computational model when we model it in vpython. Here is a sample code that creates a vector from a baseball to a tennis ball. Notice that the tail begins from the baseball and the head points to the tennis ball. vectors

In Physics 2, you need to know that a vector always starts at a source and points to an observation location you want to get physical quantities(such as electric field, magnetic field, etc.) at. Also, you should be able to calculate the magnitude and direction of the vector in 3D space. Here is an example of a VPython model that computationally calculates such values. (The green arrow represents the position vector that starts from the source, which is a proton(red ball), to the arbitrary observation location.) (Click Run on the upper left corner in order to display the model.) 3D Vector

Examples

Simple

Which of the following statements is correct? (Can be more than one)

1. [math]\displaystyle{ \overrightarrow{c} = \overrightarrow{a} + \overrightarrow{b} }[/math]

2. [math]\displaystyle{ \overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c} }[/math]

3. [math]\displaystyle{ \overrightarrow{a} = \overrightarrow{c} + \overrightarrow{b} }[/math]

4. [math]\displaystyle{ \overrightarrow{b} = \overrightarrow{c} + \overrightarrow{a} }[/math]

The answer is option number 2 and option number 4.

Middling

1. What is the magnitude of the vector C = A - B if A = <10, 5, 8> and B = <9, 4, 3>?

First we need to find the vector C: A - B = [math]\displaystyle{ \lt (10-9), (5-4), (8-3)\gt = \lt 1, 1, 5\gt = C }[/math]

[math]\displaystyle{ \sqrt{(1)^2 + 1^2 + 5^2} = 5.196 }[/math]

2. What is the cross product of A = <1,2,3> and B = <9,4,5>?

Use the equation for cross product: a x b= <a1, a2, a3> x <b1, b2, b3> = <a2*b3 - a3*b2> i - <a1*b3 - a3*b1> j + <a1*b2 - a2*b1> k

A x B[math]\displaystyle{ =\lt 2*5 - 3*4, 1*5 - 3*9, 1*4 - 2*9\gt = \lt -2, -22, -14\gt }[/math]

Difficult

What is the unit vector in the direction of the vector <10, 5, 8>? First you have to find the magnitude of the vector given: [math]\displaystyle{ \sqrt{10^2 + (5)^2 + 8^2} = 13.74 }[/math]

Finally divide the vector by its magnitude to get the unit vector:

[math]\displaystyle{ \tfrac{\lt 10, 5, 8\gt }{13.74} }[/math]

= <.727, .364, .582> Notice that the magnitude of the unit vector is equal to 1

Connectedness

1.Vectors will be used in many applications in physics. These can be two dimenstional or three dimensional vectors. Vectors are used to represent forces, fields and momentum.

2.Vectors has been used in many application problems in engineering majors. In engineering applications, vectors are used to a lot of quantities which have both magnitude and direction. For example in Biomedcial Engineering applications, vectors are used to represent the velocity of a flow to further calculate the flow rate and some other related quantities.

History

The discovery and use of vectors can date back to the ancient philosophers, Aristotle and Heron. The theory can also be found in the first article of Newtons Principia Mathematica. In the early 19th century Caspar Wessel, Jean Robert Argand, Carl Friedrich Gauss, and a few more depicted and worked with complex numbers as points on a 2D plane. in 1827, August Ferdinand published a book introducing line segments labelled with letters. he wrote about vectors without the name "vector".

In 1835, Giusto Bellavitis abstracted the basic idea of a vector while establishing the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent (meaning equal length). He found a relationship and created the first set of vectors. Also in 1835 Hamilton founded "quaternions", which were 4D planes and equations with vectors.

William Rowan Hamilton devised the name "vector" as part of his system of quaternions consisting of three dimensional vectors.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today.

See also

External links

Here is a link on more mathematical computations on vectors: [1]

Further Reading

Vector Analysis by Josiah Willard Gibbs

Introduction to Matrices and Vectors by Jacob T. Schwartz

References

https://www.mathsisfun.com/algebra/vectors.html

http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf

[2]