Resolution of a Vector in a Plane: Vertical and Horizontal Components

All physical quantities, such as force, momentum, velocity, and acceleration, are vector quantities, meaning that they have both a magnitude and a direction associated with them. The vector is represented as an arrow-headed line, with the tip of the arrow representing the head and the line representing the tail.

Consider the following scenario: there are two pathways, denoted by the letters i and j, where i and j represent the horizontal and vertical components of a vector, respectively. As a result, a Pythagoras theorem from the following figure may be used to compute the displacement:

Consider the following

Horizontal component (i) is 12m 

Vertical component (j) is 5m

122+ 52 = 144 +25=169 = 13

As a result, the displacement is 13 metres.

Let us now define the horizontal and vertical components.

Definition of Horizontal Component

In science, the horizontal component of a force is defined as the portion of the force that moves parallel to the horizontal axis.

Let us say that you kick a football. The force of the kick can be broken down into two parts: a horizontal component that moves the football along the ground, and a vertical component that moves the football at an angle to the ground.

Definition of Vertical Component

When we talk about the vertical component of a vector, we are referring to the part or component of the vector that is perpendicular to a horizontal or level plane.

How to get a vector’s rectangular components?

First, we need to know that vectors are made up of two rectangular parts, the horizontal and vertical parts. The horizontal part is on the x-axis, and the vertical part is on the y-axis. This helps us understand vectors.

Consider the following scenario: if the light were coming from above, the horizontal component would look like a shadow of the vector r descending along the x axis.

Additionally, if the light is coming from one side, the vertical component will resemble the shadow of vector r on the  y-axis

The vertical component will be denoted as rv

The horizontal component will be denoted as rh

Let us call the angle made by the vector   r   with the horizontal component as θ.

If we look closely, we can see the three vectors rv , rh   and θ,

we can state that the three sides of a right angled triangle are formed.

So, based on trigonometry, we may say, 

  rv  = |  r| cos

As a result, because θ r is the hypotenuse and rh is the adjacent side, adj/hyp = cosine of the angle; as a result, we can estimate the magnitude of the vector on the horizontal plane by using this approach given, the magnitude of the vector r and the angle it  makes with the vector on the horizontal plane

As an example, because the vertical component resembles the opposite side of the triangle and because the ratio of opp/hyp = sine of the angle is given by, the size of the vertical component can be calculated as

  rh  = |  r| sin

Assuming that we already know how to compute the magnitudes of both rectangular components of the two vectors, we must now determine the direction and magnitude of the final vector, based on what we have learned about calculating their horizontal and vertical components so far. This may be performed quickly and easily through the use of a graphical tool, which we shall illustrate in the following section.

For example, if the horizontal and vertical components have magnitudes of 100 cm and 40 cm respectively, we can use the information we have been given to create a right-angled triangle. In order to understand how the vectors behave, we must represent the lengths of the vectors in a manner that is proportionate to their magnitude. i.e.

The hypotenuse of the resultant vector may now be drawn, and the vector length provides us with information on the magnitude and direction of the resultant vector, respectively.