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Maths > Vector Algebra > Types of Vectors
Vector Algebra

Types of Vectors

In this article, we will look at different types of vectors like zero, unit, coinitial, collinear, equal and negative vectors. Further, we will solve some examples to get a better understanding.

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Types of Vectors

Zero Vector

We know that all vectors have initial and terminal points. A Zero vector or a null vector is one in which these two points coincide. It is denoted as \( \vec{0} \). Since the magnitude is zero, we cannot assign a direction to these vectors. Alternatively, zero vectors can have any direction. Some examples of zero vectors are \( \vec{AA} \), \( \vec{BB} \), etc.

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Unit Vector

A Unit vector is a vector having a magnitude of unity or 1 unit. A unit vector in the direction of a given vector \( \vec{a} \) is denoted as \( \hat{a} \).

Coinitial Vectors

Coinitial vectors are two or more vectors which have the same initial point. For example, \( \vec{AB} \) and \( \vec{AC} \) are coinitial vectors since they have the same initial point ‘A’.

Collinear Vectors

Collinear vectors are two or more vectors which are parallel to the same line irrespective of their magnitudes and direction.

Equal Vectors

If two vectors \( \vec{a} \) and \( \vec{b} \) have the same magnitude and direction regardless of the positions of their initial points, then they are Equal vectors. These vectors are written as \( \vec{a} \) = \( \vec{b} \).

Negative of a Vector

Let’s say that there is a vector \( \vec{AB} \) having a certain magnitude and direction. Now, if there is a vector whose magnitude is same as that of vector \( \vec{AB} \) but the direction is opposite, then it is called negative of the given vector \( \vec{AB} \). For example, vector \( \vec{BA} \) is the negative of vector \( \vec{AB} \). It is written as \( \vec{BA} \) = – \( \vec{AB} \).

Important Note: In this article, we will be talking only about free vectors. Free vectors are ones which can be subjected to parallel displacements without changing their magnitudes and direction.

Vector Algebra

Now, let us look at an example to understand the different types of vectors.

Example 1

In the figure given below, identify Collinear, Equal and Coinitial vectors:

types of vectors

Solution: By definition, we know that

  • Collinear vectors are two or more vectors parallel to the same line irrespective of their magnitudes and direction. Hence, in the given figure, the following vectors are collinear: \( \vec{a} \), \( \vec{c} \), and \( \vec{d} \).
  • Equal vectors have the same magnitudes and direction regardless of their initial points. Hence, in the given figure, the following vectors are equal: \( \vec{a} \) and \( \vec{c} \).
  • Coinitial vectors are two or more vectors having the same initial point. Hence, in the given figure, the following vectors are coinitial: \( \vec{b} \), \( \vec{c} \), and \( \vec{d} \).

More Solved Examples for You

Question 1: In the given figure, identify the following vectors

  1. Coinitial
  2. Equal
  3. Collinear but not equal

types of vectors

Answer :

  • Coinitial vectors have the same initial point. In the figure given above, vectors \( \vec{a} \) and \( \vec{d} \) are the two vectors which have the same initial point P.
  • Equal vectors have same magnitudes and direction. In the figure given above, vectors \( \vec{b} \) and \( \vec{d} \) are equal vectors.
  • Collinear vectors are two or more vectors parallel to the same line. In the figure given above, vectors \( \vec{a} \) and \( \vec{c} \) are parallel and hence, collinear. Also, vectors \( \vec{b} \) and \( \vec{d} \) are parallel and hence, collinear. We know that vectors \( \vec{b} \) and \( \vec{d} \) are also equal. Hence, vectors \( \vec{a} \) and \( \vec{c} \) are collinear but not equal.

Question 2: What is the formula for vector?

Answer: We can define the magnitude of vector →PQ, which is the distance between the initial point P and the endpoint Q. Furthermore, in symbols, the magnitude of →PQ is written as | →PQ |. Most importantly, if the coordinates of the initial point and the endpoint of a vector are given then we can use the distance formula to find its magnitude.

Question 3: What is a negative vector?

Answer: It refers to a vector that points in the opposite direction to the reference positive direction. In simple words, it is a vector that has the opposite direction to the reference positive direction. In addition, similar to a scalar, vectors can also be subtracted or added.

Question 4: Is acceleration a vector?

Answer: Yes, acceleration is a vector quantity because it has both direction and magnitude. Furthermore, a negative acceleration means that the object is slowing down. And it occurs in the opposite direction as the movement of the object.

Question 5: Are parallel vectors collinear?

Answer: These are those vectors that have the same or parallel support. In addition, they can have equal or unequal magnitudes and their directions can be opposite or same. Most importantly, two vectors are collinear if they have the same direction or are parallel or anti-parallel.

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