Types of vectors

A vector is a physical quantity that has a magnitudeLength of a vector. and a direction. In various areas of physics and mathematics, vectors are used to understand the behavior of directional quantities in two and three dimensional spaces.

In this Answer, we will discuss various types of vectors.

1) Unit vector

A unit vector is a vector with a magnitude of one. Any vector can become a unit vector by dividing it by the magnitude of the given vector. Suppose we have a vector$\vec{v}$, we represent its unit vector$\hat{v}$(pronounced as v-hat) as:

Here,$|v|$represents the magnitude of the vector.

The following diagram helps us understand the unit vectors in three dimensions.

We represent the unit vector in the x-axis, y-axis, and z-axis with $\hat{i}$, $\hat{j}$, and $\hat{k}$ respectively.

Example

Suppose we have a vector $\vec{v}$ such that:
$\vec{v} = (3,4)$

• Let's calculate its magnitude $|v|$:

• We find the unit vector$v̂$using the formula we have studied above:

• Now we find the magnitude of this unit vector$|v̂|$:

Hence, we prove the magnitude of a unit vector is 1.

2) Zero vector

A zero vector is a vector with a magnitude of zero. This means that the head and tail of the vector lie on the same point. We represent a zero vector as $\vec{O}$ where:

• For 2D space:

• For 3D space:

Note: A zero vector has no direction.

3) Displacement vector

A vector that gives the displacementThe action of moving something from its place or position between two points. Suppose we have a vector $\vec{v}$, whose initial point is $P1(x1,y1)$ and ending point is $P2(x2,y2)$, we represent the displacement vector using the $d$ symbol:

Let's represent it graphically:

Example

Suppose we have a vector $\vec{v}$ with starting point $P1(5,3)$ and ending point $P2 (8,6)$. To represent the displacement vector, we write:

This vector indicates that we have moved 3 units to the right (in the x-direction) and 5 units upwards (in the y-direction) from point $P1$ to point $P2$.

4) Position vector

A vector that gives information about the position of a vector with respect to originAll coordinate values equals 0.. Suppose we have a vector $\vec{v}$, whose starting point is $O(0,0)$(origin) and ending point is $P(x,y)$, we represent the position vector using the $r$ symbol:

Let's represent it graphically:

Example

Suppose we have a point $P$$(5,3$). To represent the position vector from the origin $O(0,0)$ to point $P$, we write:

This position vector indicates that point $P$ is located $5$ units to the right (positive x-direction) and $3$ units up (positive y-direction) from the origin.

5) Negative vector

A vector in the opposite direction but having same magnitude as the original vector. Suppose we have a vector $\vec{v}$. Its negative vector can be represented graphically as:

Example

Suppose we have a vector $\vec{v}1(3,6)$, the negative of this vector will be $\vec{-v1}(-3,-6)$.

Note: The coordinates of a negative vector are opposite in sign from the corresponding coordinates of the original vector.

6) Co-initial vectors

Vectors that have the same starting point. Suppose we have two vectors: $\vec{v}1$ and $\vec{v}2$. Vector $\vec{v}1$ has starting point at $(x1,y1)$and ending point at $(x2,y2)$.Vector $\vec{v}2$has starting point at $(a1,b1)$and ending point at $(a2,b2)$. These two vectors will be co-initial vectors if $(x1,y1)$ is equal to $(a1,b1)$. Let's represent it graphically:

Example

Suppose we have two vectors $\vec{v}1$ and $\vec{v}2$ that both start at the point $(2, 3)$ and end at points $(6, 5)$ and $(4, 8)$ respectively. Both vectors start from the same point, so they are co-initial vectors.

7) Like vectors

Vectors that have same the direction but not necessarily the same magnitude. Suppose we have two like vectors: $\vec{v}1$and $\vec{v}2$. They can be represented graphically as:

Example

Suppose we have two vectors $\vec{v}1(3,0)$ and $\vec{v}2(5,0)$. These vectors are in a positive x-axis direction, so they are like vectors as they have the same direction.

8) Unlike vectors

Vectors that have different direction but not necessarily the same magnitude. Suppose we have two unlike vectors: $\vec{v}1$ and $\vec{v}2$. They can be represented graphically as:

Example

Suppose we have two vectors $\vec{v}1(3,0)$ and $\vec{v}2(0,5)$. These vectors are unlike vectors as they have different direction: $\vec{v}1$is along the x-axis, while $\vec{v}2$ is along the y-axis.

9) Co-planar vectors

Vectors that lie in the same plane. This means they can be positioned on the same surface without any of them extending out of that plane. Suppose we have two co-planar vectors: $\vec{v}1$ and $\vec{v}2$. They can be represented graphically as:

Note: In a 2D space, any two vectors are always co-planar because they lie on the same flat plane.

Example

Suppose we have two vectors $\vec{v}1(3,1)$ and $\vec{v}2(-2,4)$. These vectors are co-planar vectors as they both lie in the same flat plane.

10) Co-linear vectors

Vectors that are linear or parallel to each other. They can also have opposite directions, but they have to be parallel. They are also known as parallel vectors. Suppose we have two co-linear vectors: $\vec{v}1$ and $\vec{v}2$. They can be represented graphically as:

Example

Suppose we have two vectors $\vec{v}1(3,6)$ and $\vec{v}2(6,12)$. These vectors are co-linear vectors as they are parallel to each other.

Note: Col-linear vectors are scalar multiples of each other, meaning one can be expressed as a linear combination of the other. The angle between these vector is $0^o$.

11) Equal vectors

Vectors that have same direction and magnitude. Suppose we have two equal vectors: $\vec{v}1$ and $\vec{v}2$. They can be represented graphically as:

Example

Suppose we have two vectors $\vec{v}1(3,6)$ and $\vec{v}2(3,6)$. These vectors are equal vectors as they have the same direction and the same magnitude.

Note: The x and y coordinates for two equal vectors are always same.

12) Orthogonal vectors

The vectors that are perpendicular to each other and have an angle of $90^o$ between them. Suppose we have two orthogonal vectors: $\vec{v}1$ and $\vec{v}2$. They can be represented graphically as:

Example

Suppose we have two vectors $\vec{v}1(3,0)$ and $\vec{v}2(0,-4)$. These vectors are orthogonal as they are perpendicular to each other: $\vec{v}1$is along the x-axis, while $\vec{v}2$ is along the y-axis.

Conclusion

Vectors play a vital role in understanding the movement of objects around us. Each type of vector serves a distinct purpose. These types enable us to describe various relationships and phenomena in mathematics, physics, and various other disciplines of science.