Resolution of Vector Rectangular Components

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Resolution of a Vector in a Plane

All Physical quantities like force, momentum, velocity, acceleration are all vector quantities because they have both magnitude and direction. We represent the vector as an arrow-headed line, where the tip of the arrow is the head and the line is the tail.


Let’s suppose there are two paths, viz: A and B, where A and B are horizontal and vertical components of a vector, respectively. So, the displacement can be calculated by using a Pythagoras theorem from the following diagram:

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\[\sqrt{3^{2}}\] + \[\sqrt{4^{2}}\] = \[\sqrt{9}\] + \[\sqrt{16}\] = \[\sqrt{25}\] = 5    

So,  5 m is displacement.

Now, let’s discuss what is the horizontal and vertical component.


Horizontal Component Definition

In science, we define the horizontal component of a force as the part of the force that moves directly in a line parallel to the horizontal axis. 


Let’s suppose that you kick a football, so now, the force of the kick can be divided into a horizontal component, which is moving the football parallel to the ground, and a vertical component that moves the football at a right angle to the surface/ground.


Vertical Component Definition

We define the vertical component as that part or a component of a vector that lies perpendicular to a horizontal or level plane.


Resolution of a Vector

Resolution of a vector is the splitting of a single vector into two or more vectors in different directions which together produce a similar effect as is produced by a single vector itself. The vectors formed after splitting are called component vectors.

Let’s understand this with the following diagram:

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Here, 

OB\[^{\rightarrow}\] = ax =  vector along the x-axis (It is the horizontal component formula)

OD\[^{\rightarrow}\] = ay = vector along the y-axis (It s the vertical component formula)

From here, we obtained the horizontal and vertical components of a vector, which is a vector a\[^{\rightarrow}\].

From the triangle law of addition, we can use the formula as:

OC\[^{\rightarrow}\] = OB\[^{\rightarrow}\] + OD\[^{\rightarrow}\]

a\[^{\rightarrow}\] = ax +  ay….(1)

Here, we can see that OCB is right-angled, so using the formula of the trigonometric function, we get the angular components along the x and y-axis, respectively:

Since

OB/OC = Cos

OB = OC Cos

So, 

ax = a\[^{\rightarrow}\] Cos….(2)

Similarly, 

BC/OC = Sin

ay = a\[^{\rightarrow}\] Sin….(3)

Now, eq (3) ÷ eq (2), we get the tangent of component, which is given by:

(a\[^{\rightarrow}\] SinΘ)/a\[^{\rightarrow}\] CosΘ() = ay/ ax 

So,

tan = BC/OB = ay/ ax ….(4)


Rectangular Components of Vectors in Three Dimensions

We define rectangular components of vectors in Three Dimensions in the following manner:

If the coordinates of a point P, i.e., x, y, and z, the vector joining point P to the origin is called the position vector. The position vector of point P is equal to the sum of these coordinates, which is given by:

         x + y + z


Rectangular Components 

Rectangular components of a vector in three dimensions can be better understood by going through the following context:

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Let’s suppose that vector A\[^{\rightarrow}\] is presented by the vector OR\[^{\rightarrow}\]. Now, taking O as the origin and construct a rectangular parallelopiped with its three edges along with the three rectangular axes, viz: X, Y, and Z. Here, we can notice that A\[^{\rightarrow}\] represents the diagonal of the rectangular parallelopiped whose intercepts are the ax,  ax,  and ax, respectively. We call these intercepts the three rectangular components of A\[^{\rightarrow}\]. 

Now, using the triangular law of vector addition, we have:

OR\[^{\rightarrow}\]  + OT\[^{\rightarrow}\]  + TR\[^{\rightarrow}\] 

Using the parallel law of vector addition, we have:

OT\[^{\rightarrow}\] = OS\[^{\rightarrow}\] + OP\[^{\rightarrow}\]

OR\[^{\rightarrow}\] = (OS\[^{\rightarrow}\] + OP\[^{\rightarrow}\]) + TR\[^{\rightarrow}\] ……(5)

Here, one must notice that TR\[^{\rightarrow}\] = OQ\[^{\rightarrow}\]. So, rewriting equation (5) in the following manner:

OR\[^{\rightarrow}\] = (OS\[^{\rightarrow}\] + OP\[^{\rightarrow}\]) + OQ\[^{\rightarrow}\]

Or,

A\[^{\rightarrow}\] = A\[^{\rightarrow}\]z + A\[^{\rightarrow}\]x + A\[^{\rightarrow}\]y = A\[^{\rightarrow}\]x + A\[^{\rightarrow}\]y + A\[^{\rightarrow}\]z ……(6)

Therefore,

A\[^{\rightarrow}\] = Ai\[^{\rightarrow}\]x + Aj\[^{\rightarrow}\]y + Ak\[^{\rightarrow}\]z

Also, 

OR² = OT² + TR²

OP² + OS² + TR²

Now,

A² = A²x + A²y + A²z ……..(7)


Resolution of Rectangular Vectors in Three Dimensions in their Direction Cosines 

Now, we can restate equation (6) in the following manner:

A = \[\sqrt{A^{2}x + A^{2}y + A^{2}z}\]

If α, β, and γ are the angles which the vector A\[^{\rightarrow}\] makes with the X, Y, and Z-axis, respectively, then we have:

Cos α = Ax\[^{\rightarrow}\] / A\[^{\rightarrow}\] ⇒ Ax\[^{\rightarrow}\] = A\[^{\rightarrow}\] Cos α …..(a)

Cos β = A\[^{\rightarrow}\]y / A\[^{\rightarrow}\] ⇒ A\[^{\rightarrow}\]y = A\[^{\rightarrow}\] Cos β ….(b)

Cos γ = Az\[^{\rightarrow}\] / A\[^{\rightarrow}\] ⇒ Az\[^{\rightarrow}\] = A\[^{\rightarrow}\] Cos γ …..(c)

We must note that Cos α,  Cos β, and Cos γ are direction cosines of vectors Ax\[^{\rightarrow}\], Ay\[^{\rightarrow}\], and Az\[^{\rightarrow}\], respectively.

Now, putting the values of equations (a), (b), and © in the equation (7), we get:

A² = A² Cos²α + A²Cos² β + A² Cos² γ …….(8)

So, we get the equation as:

Cos² α + Cos² β + Cos² γ = 1 

Here, we conclude that the squares of the direction cosines of three vectors are always constant, i.e., unity.

FAQ (Frequently Asked Questions)

Question 1: How Do We Calculate the Horizontal and Vertical Components of Vectors?

Answer: We know that the velocity or the force that is parallel to the horizontal axis is called the horizontal component and the quantity that is parallel to the vertical axis is called the vertical component. We can calculate the horizontal and vertical components by representing these on the right-angled triangle. The hypotenuse is the force or velocity and the angle is used to determine/calculate the vertical and horizontal components of vectors.

Question 2: What Do You Mean by the Resolution of Vectors?

Answer: We can split a vector in a vector space into its two components, viz: horizontal and vertical components.


Where the horizontal component is a product of the magnitude of the vector and the cosine of the horizontal angle, i.e., the angle that falls on the horizontal axis, while the vertical component of a vector is the product of the magnitude of the vector and the sine of its horizontal angle (angle along the horizontal axis).

Question 3: State the Definition of a Vector.

Answer: A vector is a physical quantity that has both magnitude and direction. A vector quantity describes the movement of moving objects from one point to another.

Question 4: Can a Vector be Resolved?

Answer: Yes.

Let’s consider two numbers, say 5 and 8, which are is added to get 13. Furthermore, 13 is split or resolved.