Resolution of a Vector in Plane-Rectangular Components

A vector can be defined as a physical quantity with both magnitude and direction. It can be resolved or split into parts known as its ‘components’:  magnitude and direction. An arrow represents vectors. Some important terms related to the concept are as follows:

• Magnitude: All vectors have a certain length. Magnitude is defined as the length of a vector.
• Components: A vector is split into parts, known as its components. In 2 – D, the components are the horizontal components and vertical components, whereas, in the 3 – D analysis, the components are represented as x, y and z.
• Coordinates: These are the numbers representing a position on the axis. For example, the x coordinate represents the position relative to the x-axis.
• Scalar: It is a physical quantity having magnitude only.
• Unit vector: It is a vector having a magnitude of 1.
• Velocity is defined as the rate of change of displacement at a given time.
• Acceleration is the rate of change in velocity at a given point in time.
• Displacement defines the direction and distance between two objects in a straight line.

Vectors are used to represent physical quantities like acceleration, displacement and velocity. The vectors are represented by an arrow pointing towards the direction of the quantity. This helps in analysing the direction and the length defines the magnitude of the vector.

Horizontal Component and Vertical Component 

The horizontal component is defined as the part of the force that acts in parallel in the direction of the horizontal line. The vertical component is defined as the vector lying perpendicular to the horizon.

When the two components of a vector are together, they form a right angle. 

In two dimensions, an object is subjected to a further divided force into two components. These horizontal and vertical components of vectors help analyse the strength, heat and speed required to obtain the desired results.

Let’s consider an example of a person skiing on ice. Here, the slope of the mountain acts as a horizontal component and the person as a vertical component.

Another example is of a person lifting a box. Here, the person’s body is considered a resultant vector that can be split into horizontal and vertical components. Thus, we can easily find the magnitude and direction of the horizontal and vertical components from analysing the components.

Applications of the Horizontal Component of a Vector

• It is used to determine the effect of wind on the plane’s speed in aerodynamics.
• It is used in towboats in the ocean when the towboat is moving in the same direction as the ocean waves.
• It is used by drivers to decide the best possible use of speed, altitude, etc.

Resolution of a Vector

If the components of a given vector are in a perpendicular direction to each other, they are said to be rectangular components of a vector. The term resolution of a vector means resolving a vector into its components that define the effect of a vector in a given direction. 

For example, consider a vector (4,1). The vector’s vertical component is 1 (y-axis) and the horizontal component is 4 (x-axis).

To better understand this, consider an example of a ball rolling on a flat table at 15 degrees with a speed of 7 m/sec at a direction parallel to the edge. We need to find out the time required by the ball to roll the edge of 1 metre to the right.

The speed of the ball is the magnitude of the velocity vector, so velocity vector = V;

Now, to find out the velocity of a ball, we require the x and y components of the ball. So the velocity vector of the x component can be written as vx and the y component is written as vy

So V = (Vx, Vy)

Now, as we know, θ = 15 °

and horizontal component Vx = Vcosθ

vertical component Vy = Vsinθ  

sinθ = Vy/V and cosθ = Vx/V

By multiplying both the equations by V on both sides, we will get:

Vx = vcosθ and

Vy = vsinθ

Now, we know, tanθ = sinθ /cosθ;

so, Vx = Vcosθ = Vy / tanθ

      Vy = Vsinθ = Vxtanθ

      V = Vy / sinθ = Vx / cosθ

Also, we know that

Vx = Vcosθ

By putting the given value of Vx = 7 m/s and the value of θ = 15°, we get:

Vx = Vcosθ 

      = (7 m/s)cos 15°

      = 6.8 m/s

Now, we know that the ball is travelling at 6.8 m/s.

Finding out the time required to travel 1-metre edge:

   1.0 m / 6.8 m

= 0.15s

So, if the ball is going in the direction of x-components, it will require a time of 0.15 seconds.

Similarly, for the y component,

Vy = sinθ

     = (7 m/s)sin 15°

     = 1.8 m/s

Addition and Subtraction of Vectors

A vector having a length of 5 will have a horizontal component of 4 units and a vertical component of 3 units at 36.9°. If we want to add a new vector of the same length, it is done as follows:

Add the two horizontal components: 4+4 = 8

and the two vertical components: 3+3 = 6

Now, the resultant vector is calculated by placing the tail of the vertical component to the head of the horizontal component and drawing a line. This new line is the resultant vector.

To simply subtract the vectors, the two horizontal components are subtracted from each other and two vertical components are subtracted from each other and then the resultant vector is plotted.

Conclusion

Vectors can be resolved or split into two components: vertical component and horizontal component. Vectors are represented by arrows that define the magnitude (the length) and direction of a quantity. The quantities having both magnitude and direction are called vector quantities, whereas those having only magnitude are referred to as scalar quantities. 

Vectors are analysed by breaking them into their components. When the analysis is done in two dimensions, the components are:

• Horizontal component 

• Vertical component

And when the analysis is done in three dimensions, the components are:

• x 
• y 
• z

Vectors can be subtracted or added by arranging them end-to-end on axes.