Vector

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 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.

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

For vectors and and numbers and ,

The following equalities hold for all vectors and scalars:

• (Commutivity of vector addition)

• (Associativity of vector addition)

• (Distributivity of scalars)

• (Associativity of scalar multiplication)

• the existence of a zero vector (additive identity), , such that

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

Vectors in this form are added by components:

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

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