What is vector in math?

A vector is an object that has both a magnitude and a direction. Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector and with an arrow indicating the direction. The direction of the vector is from its tail to its head.

Two vectors are the same if they have the same magnitude and direction. This means that if we take a vector and translate it to a new position (without rotating it), then the vector we obtain at the end of this process is the same vector we had in the beginning.

Two examples of vectors are those that represent force and velocity. Both force and velocity are in a particular direction. The magnitude of the vector would indicate the strength of the force or the speed associated with the velocity.

We denote vectors using boldface as in $\vc{a}$ or $\vc{b}$. Especially when writing by hand where one cannot easily write in boldface, people will sometimes denote vectors using arrows as in $\vec{a}$ or $\vec{b}$, or they use other markings. We won't need to use arrows here. We denote the magnitude of the vector $\vc{a}$ by $\|\vc{a}\|$. When we want to refer to a number and stress that it is not a vector, we can call the number a scalar. We will denote scalars with italics, as in $a$ or $b$.

You can explore the concept of the magnitude and direction of a vector using the below applet. Note that moving the vector around doesn't change the vector, as the position of the vector doesn't affect the magnitude or the direction. But if you stretch or turn the vector by moving just its head or its tail, the magnitude or direction will change. (This applet also shows the coordinates of the vector, which you can read about in another page.)

There is one important exception to vectors having a direction. The zero vector, denoted by a boldface $\vc{0}$, is the vector of zero length. Since it has no length, it is not pointing in any particular direction. There is only one vector of zero length, so we can speak of the zero vector.

We can define a number of operations on vectors geometrically without reference to any coordinate system. Here we define addition, subtraction, and multiplication by a scalar. On separate pages, we discuss two different ways to multiply two vectors together: the dot product and the cross product.

Given two vectors $\vc{a}$ and $\vc{b}$, we form their sum $\vc{a}+\vc{b}$, as follows. We translate the vector $\vc{b}$ until its tail coincides with the head of $\vc{a}$. (Recall such translation does not change a vector.) Then, the directed line segment from the tail of $\vc{a}$ to the head of $\vc{b}$ is the vector $\vc{a}+\vc{b}$.

The vector addition is the way forces and velocities combine. For example, if a car is travelling due north at 20 miles per hour and a child in the back seat behind the driver throws an object at 20 miles per hour toward his sibling who is sitting due east of him, then the velocity of the object (relative to the ground!) will be in a north-easterly direction. The velocity vectors form a right triangle, where the total velocity is the hypotenuse. Therefore, the total speed of the object (i.e., the magnitude of the velocity vector) is $\sqrt{20^2+20^2}=20\sqrt{2}$ miles per hour relative to the ground.

Addition of vectors satisfies two important properties.

You can explore the properties of vector addition with the following applet. (This applet also shows the coordinates of the vectors, which you can read about in another page.)

Before we define subtraction, we define the vector $-\vc{a}$, which is the opposite of $\vc{a}$. The vector $-\vc{a}$ is the vector with the same magnitude as $\vc{a}$ but that is pointed in the opposite direction.

We define subtraction as addition with the opposite of a vector: $$\vc{b}-\vc{a} = \vc{b} + (-\vc{a}).$$ This is equivalent to turning vector $\vc{a}$ around in the applying the above rules for addition. Can you see how the vector $\vc{x}$ in the below figure is equal to $\vc{b}-\vc{a}$? Notice how this is the same as stating that $\vc{a}+\vc{x}=\vc{b}$, just like with subtraction of scalar numbers.

Given a vector $\vc{a}$ and a real number (scalar) $\lambda$, we can form the vector $\lambda\vc{a}$ as follows. If $\lambda$ is positive, then $\lambda\vc{a}$ is the vector whose direction is the same as the direction of $\vc{a}$ and whose length is $\lambda$ times the length of $\vc{a}$. In this case, multiplication by $\lambda$ simply stretches (if $\lambda>1$) or compresses (if $0 < \lambda <1$) the vector $\vc{a}$.

If, on the other hand, $\lambda$ is negative, then we have to take the opposite of $\vc{a}$ before stretching or compressing it. In other words, the vector $\lambda\vc{a}$ points in the opposite direction of $\vc{a}$, and the length of $\lambda\vc{a}$ is $|\lambda|$ times the length of $\vc{a}$. No matter the sign of $\lambda$, we observe that the magnitude of $\lambda\vc{a}$ is $|\lambda|$ times the magnitude of $\vc{a}$: $\| \lambda \vc{a}\| = |\lambda| \|\vc{a}\|$.

Scalar multiplications satisfies many of the same properties as the usual multiplication.

In the last formula, the zero on the left is the number 0, while the zero on the right is the vector $\vc{0}$, which is the unique vector whose length is zero.

In mathematics and physics, vector is a term that refers colloquially to some quantities that cannot be expressed by a single number (a scalar), or to elements of some vector spaces.

Historically, vectors were introduced in geometry and physics (typically in mechanics) for quantities that have both a magnitude and a direction, such as displacements, forces and velocity. Such quantities are represented by geometric vectors in the same way as distances, masses and time are represented by real numbers.

The term vector is also used, in some contexts, for tuples, which are finite sequences of numbers of a fixed length.

Both geometric vectors and tuples can be added and scaled, and these vector operations led to the concept of a vector space, which is a set equipped with a vector addition and a scalar multiplication that satisfy some axioms generalizing the main properties of operations on the above sorts of vectors. A vector space formed by geometric vectors is called a Euclidean vector space, and a vector space formed by tuples is called a coordinate vector space.

Many vector spaces are considered in mathematics, such as extension field, polynomial rings, algebras and function spaces. The term vector is generally not used for elements of these vectors spaces, and is generally reserved for geometric vectors, tuples, and elements of unspecified vector spaces (for example, when discussing general properties of vector spaces).

Every algebra over a field is a vector space, but elements of an algebra are generally not called vectors. However, in some cases, they are called vectors, mainly due to historical reasons.

The set R n {\displaystyle \mathbb {R} ^{n}} of tuples of n real numbers has a natural structure of vector space defined by component-wise addition and scalar multiplication. It is common to call these tuples vectors, even in contexts where vector-space operations do not apply. More generally, when some data can be represented naturally by vectors, they are often called vectors even when addition and scalar multiplication of vectors are not valid operations on these data.[disputed – discuss] Here are some examples.

A vector field is a vector-valued function that, generally, has a domain of the same dimension (as a manifold) as its codomain,

A vector is a quantity or phenomenon that has two independent properties: magnitude and direction. The term also denotes the mathematical or geometrical representation of such a quantity. Examples of vectors in nature are velocity, momentum, force, electromagnetic fields, and weight.

• Zero Vector.
• Unit Vector.
• Position Vector.
• Co-initial Vector.
• Like and Unlike Vectors.
• Co-planar Vector.
• Collinear Vector.
• Equal Vector.