Vector

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Vectors are quantities that are specified by both a direction and magnitude of length. The way you draw a vector is by drawing an arrow. This arrow has a length and points in some direction.

Vectora.jpg


When we use vectors we need to introduce the coordinate system. We represent vectors in terms of their components along the coordinate axis. So if we have an arrow we can now communicate with someone else about how much the vector goes in the x direction, the y direction and the z direction. When we have a vector quantity we put an arrow on top of the labeling letter to remind us that it is a vector. It looks like so \vec A.

A vector is a combination of unit vectors that are placed along the the coordinate axes. These unit vector have length one. A unit vector is denoted by a lowercase letter with a hat above it, like so, \hat{a}. It is pronounced as a-hat. Our standard correspond to the x-y-z coordinate system. \hat{i} points along the x-axis \hat{j} points along the y-axis and \hat{k} points along the z-axis. We can express any vector in terms of its components relying on the unit vectors. With \left \langle a_1, a_2, a_3 \right \rangle as any numerical values we can write any vector in the following form \vec A = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}. The other notation for a vector  \vec A = \left \langle a_1, a_2, a_3 \right \rangle

Unitvec.jpg


If we have two points in space we can connect them and make it into a vector. example in 2d Another vector is \vec A = 3 \hat{i} + 3 \hat{j} + 2 \hat{k}. Vector A is 3 units in the x and y direction and 2 units in the z direction.

VecA.jpg


If a vector is still a bit abstract to you then think of a compass. The arrow has a certain length, this is our magnitude, and it points in any direction (north, south, east, and west).


Click here for a graphical introduction to vectors:
The Euclidean vector (3, 2): Click to enlarge
'Head to tail' addition of two vectors: (3, 2) + (1,-1) = (4, 1): Click to enlarge

Perhaps the simplest vector is a Euclidean vector, represented by an arrow in Euclidean space. The length and direction of the arrow are determined by its components. Each component represents the length of the arrow in one coordinate direction. Traditionally the components of a Euclidean vector are written in the same order that an ordered pair of coordinates are written: the vector  \begin{bmatrix} 3 \\ 2\\ \end{bmatrix} has an x-component of 3 and a y-component of 2. If a vector can be expressed in components, then it can be represented by a Euclidean vector.

The length of a vector is denoted by \left| \vec A \right|. This is an actual number with only magnitude, this is a scalar quantity. Imagine that the compass isn't labeled.

As shown in the pictures, when two Euclidean vectors are added, the sum can be found by placing the two vectors 'head to tail' and finding the coordinate they reach. When a vector is multiplied by a scalar several things may occur:

  • When multiplied by a number with absolute value greater than one, the vector stretches. (Figure 1)
  • When multiplied by a number with absolute value less than one, the vector shrinks. (Figure 2)
  • When multiplied by a negative number, the vector reverses direction. (Figure 3)

Many mathematical and physical entities can be represented by vectors. For example, an object's velocity can be represented by a 'velocity vector', where each component represents the object's speed in a certain direction. Gravity can also be represented by a vector: the strength of gravity's pull corresponds to the magnitude of the vector, and the direction of the pull corresponds to the direction the vector points in. Vector Fields are another important application of vectors.

Figure 1: Stretching a vector A Figure 2: Shrinking a vector A Figure 3: Stretching and reversing the direction of a vector A
Vector bold3.PNG Vector bold4.PNG Vector bold5.PNG
Click here for an algebraic introduction to vectors:

For vectors  \vec{A},\vec{B}, and  \vec{C} and numbers d and e,

The following equalities hold for all vectors and scalars:

  •  \vec{A}+\vec{B} = \vec{B} + \vec{A} (Commutivity of vector addition)
  •  \vec{A}+(\vec{B}+\vec{C}) = (\vec{A}+\vec{B})+\vec{C}  (Associativity of vector addition)
  •  d(\vec{A}+\vec{B}) = d\vec{A} + d\vec{B} (Distributivity of scalars)
  •  d(e\vec{A}) = (de)\vec{A} (Associativity of scalar multiplication)
  • the existence of a zero vector (additive identity),  \vec{0} , such that  \vec{A} + \vec{0} = \vec{A}

The standard algebraic representation of vectors is in terms of components, although not all vectors can be expressed this way:

 \vec{A} = (a_1, a_2, ... , a_k)
 \vec{B} = (b_1, b_2, ... , b_k)

Vectors in this form are added by components:

 \vec{A} + \vec{B} = (a_1, a_2, ... , a_k)+(b_1, b_2, ... , b_k)=(a_1+b_1, a_2+b_2, ... ,a_k + b_k)

Multiplying a vector by a number, known as a scalar, means multiplying each component by that number:

 d\vec{A} = (da_1, da_2, ... , da_k)