The mathematical world is magical and fascinating. Many different quantities exist, and it’s interesting to know about them. There are two very important quantities – Scalar and Vector quantities in mathematics. Both play important roles in several mathematical applications and are important for understanding basic problems. 

Learning about vectors and their properties is very important to solve complex problems. Having a clear foundation about different concepts can pave the way for better understanding.  All these different types of vectors are real-world applications of different mathematical concepts and are used extensively.

What is a vector?

A vector is any physical quantity that has both direction and a specific magnitude. Vectors are often used to denote movement and direction. When describing the changes in position from a specific point to another, vectors are used intensively. 

They thus depict movement and a change from the initial state. These vectors are essential for several problems of mathematics and physics. They are widely used to denote different quantities and are essential to understand. 

Importance of Vectors in Everyday Life

• Used for finding direction and locating different objects. 
• Used for several mathematical and scientific inventions. 
• Used for calculating force and velocity.

Different types of vectors

Vectors are classified into several different types. The 10 most common ones are of most importance to us. 

These are-

• Unit vector
• Zero vector
• Position vector
• Collinear vector
• Coplanar vector
• Equal vector
• Displacement vector
• Co-initial vector
• Like and Unlike Vector
• Negative of vector

Unit Vector

Unit vectors are vectors whose magnitude equals 1.  Unit Vectors are usually represented by a^. All unit vectors have a length of 1. All different types of unit vectors are used to denote different types of directions. It is also important to note that any unit vector can scale off different vectors. 

All unit vectors can be used in two directions as well as 3 directions. To find any unit vector with the same direction as the given vector, you must divide the vector by its magnitude. 

Zero vector

A very special kind of vector is the zero vector. This vector has zero length. A zero vector is denoted by 0. The starting point and ending point of any zero factors coincide. 

An example of a zero vector is when a string is pulled with equal force from both directions. The zero vector is also called the null vector. The displacement of any zero vector gets nullified, and hence the resultant is zero. Thus, the null vector has no fixed direction. 

Position vector

Another very common type of vector is the position vector. A position vector depicts direction. One point is fixed, and the other point has a specific direction. The formula for finding a position vector from an arbitrary point A to point B is AB= (xk+1 – xk, yk+1 – yk). Thus the position vector starts at a point and terminates at another fixed point. 

Collinear vector

Collinear vectors are two or more vectors that are parallel to each other. This parallel characteristic is irrespective of the magnitude or direction. 

A vector is considered collinear if-

• There is an existence of n such that p vector= n x q vector. Thus, for a vector to be collinear, it should be a scalar multiple of another vector.
• Two quantities, p and q, are considered collinear vectors only if the ratios of their corresponding coordinates are always equal.
• Two vectors, p and q, are considered vectors if their cross product equals zero vector. 

Coplanar vector 

A coplanar vector lies on the same plane. These co-planar vectors lie in a 3D plane. 

There are three conditions for vectors to be co-planar-

• If any three vectors lie in an existing 3 d place and their triple scalar product is zero, it is called a Coplanar Vector.
• When three existing vectors are independent in a linear fashion in a 3d space, they are co-planar vectors.

Equal vector

Two vectors are considered equal vectors when they have the same magnitude and direction. The vectors may be equal even if they have different initial points. It is also important to note that their column vectors will also be equal if two vectors are equal. 

Displacement vector

When a position of a vector changes, it’s called a displacement vector. The displacement is measured by a shift from the initial to the final point. All the displacement vectors are always seen in the direction of displacement.

Co-initial vector

Vectors are called co-initial vectors when any two sets of vectors have the same initial point. 

Like and Unlike Vector

Two opposite sets of vectors very common in the mathematics world are Like and Unlike Vectors. Like vectors are vectors that have the same direction. Unlike vectors are vectors that have the same direction. 

Negative of vector

The negative of a vector is used to show the vector which runs opposite to the reference direction. For example, if there are two vectors A and B, vector B will be the negative of vector A. 

Operations of Vectors

Additions of Vectors

When two vectors, p and q, are present, the addition p and q are done translating the directions of each. It uses the commutative law, which shows that addition can be done in any order. 

Vectors also follow the associative law. This law states that the addition of any three vectors does not depend on which pair of vectors is added first. 

Subtraction of Vectors

The subtraction of vectors is the opposite of the addition of vectors with the same magnitude.

Conclusion

Vectors are very important, and their applications in the mathematical world are huge. A vector can be defined as any quantity that has both a magnitude and a direction. In geometric terms, a vector is a line segment. The length shows the magnitude, and the arrowhead points direction. Common examples are vectors that depict quantities like force and velocity. In both these examples, force and velocity are in a particular direction. 

Vectors are often used to denote movement and direction. When describing the changes in position from a specific point to another, vectors are used intensively.