2.3 Resolution of Vectors into Rectangular Components | Vectors | Physics 11

Introduction

Vector resolution is the process of splitting a single vector into two or more components that, when added together, produce the original vector. The most common and useful method is to break a vector down into its rectangular components — two perpendicular vectors aligned with the x and y-axes of a Cartesian coordinate system. This technique is fundamental in physics for simplifying the analysis of forces, velocities, and other vector quantities.

1. Visualizing the Components

Imagine a vector $A$ in a 2D plane, starting from the origin. This vector has a magnitude (length) $A$ and makes an angle $θ$ with the positive x-axis. We can project the "shadow" of this vector onto the x-axis and the y-axis. These projections are the rectangular components of $A$ .

x-component ( $A_{x}$ ): The projection of $A$ onto the x-axis.
y-component ( $A_{y}$ ): The projection of $A$ onto the y-axis.

Together, $A_{x}$ , $A_{y}$ , and the original vector $A$ form a right-angled triangle, with $A$ as the hypotenuse. For more details on vector properties, refer to Scalar And Vector Quantities→.

2. Calculating the Magnitudes of the Components

Using trigonometry on this right-angled triangle, we can find the magnitudes of the components, $A_{x}$ and $A_{y}$ :

x-component (Adjacent side): The magnitude of the x-component is found using the cosine function.

$A_{x} = A cos (θ)$

y-component (Opposite side): The magnitude of the y-component is found using the sine function.

$A_{y} = A sin (θ)$

Important Note: These formulas assume the angle $θ$ is measured from the positive x-axis. If the angle is given with respect to the y-axis, the roles of sine and cosine will be reversed.

3. Representing a Vector in Component Form

Once we have the components, we can express the original vector as the sum of its component vectors. Using the unit vectors $\hat{i}$ (for the x-direction) and $\hat{j}$ (for the y-direction), the vector $A$ can be written as:

$A = A_{x} + A_{y} = A_{x} \hat{i} + A_{y} \hat{j}$

Substituting the trigonometric forms, we get:

$A = (A cos θ) \hat{i} + (A sin θ) \hat{j}$

This representation is particularly useful when performing vector operations. See also Scalar Product→.

4. Reconstructing the Original Vector (Vector Composition)

The process can be reversed. If you know the components $A_{x}$ and $A_{y}$ , you can find the magnitude and direction of the original vector $A$ .

Magnitude (using Pythagorean Theorem):

$A = A_{x}^{2} + A_{y}^{2}$

Direction (using Inverse Tangent):

$θ = tan^{- 1} (\frac{A _{y}}{A _{x}})$

(Care must be taken to place the angle in the correct quadrant based on the signs of $A_{x}$ and $A_{y}$ .)

For a step-by-step process of finding a vector from its components, see Finding Vector From Its Components→.

Possible Questions and Answers

Q: Why is resolving vectors into components useful? A: It simplifies vector arithmetic, especially addition. To add multiple vectors, you can resolve each one into its x and y components, add all the x-components together to get a total x-component ( $R_{x}$ ), and add all the y-components to get a total y-component ( $R_{y}$ ). The final resultant vector can then be easily found from $R_{x}$ and $R_{y}$ .
Q: What happens to the components if the vector is in another quadrant? A: The signs of the components will change. For example, a vector in Quadrant II (angle between 90° and 180°) will have a negative x-component ( $A_{x} < 0$ ) and a positive y-component ( $A_{y} > 0$ ). The trigonometric formulas $A_{x} = A cos θ$ and $A_{y} = A sin θ$ automatically handle these signs as long as $θ$ is measured from the positive x-axis.
Q: Under what circumstances would a vector have equal magnitude components ( $A_{x} = A_{y}$ )? A: When the vector makes an angle of 45° (or 135°, 225°, 315°) with the positive x-axis. In such cases, $cos θ = sin θ$ , so $A_{x} = A cos θ = A sin θ = A_{y}$ . Graphically, this occurs when the vector bisects the angle between the x and y axes.

Q: If $A_{x} = - 2$ and $A_{y} = - 2$ , at what angle does the resultant vector make with the positive x-axis? A: Both components are negative, placing the vector in Quadrant III. The reference angle is $ϕ = tan^{- 1} (\frac{∣ - 2∣}{∣ - 2∣}) = tan^{- 1} (1) = 45°$ . Since the vector is in Quadrant III, the actual angle is $180° + 45° = 225°$ .

Summary

Vector Resolution is the process of splitting a vector into its perpendicular (rectangular) components.
The components are typically aligned with the x and y-axes.
The magnitudes of the components are found using trigonometry:
- $A_{x} = A cos θ$
- $A_{y} = A sin θ$
This technique simplifies complex vector problems, particularly vector addition, by allowing us to work with simple scalar quantities (the component magnitudes) along each axis independently.
The rectangular component method is extensively used in analyzing Equilibrium→ of forces.

Concept	Formula
x-component	$A_{x} = A cos θ$
y-component	$A_{y} = A sin θ$
Vector Magnitude	$A = A_{x}^{2} + A_{y}^{2}$
Vector Direction	$θ = tan^{- 1} (\frac{A _{y}}{A _{x}})$

Possible Questions and Answers

Q: Why is resolving vectors into components useful? A: It simplifies vector arithmetic, especially addition. To add multiple vectors, you can resolve each one into its x and y components, add all the x-components together to get a total x-component ( $R_{x}$ ), and add all the y-components to get a total y-component ( $R_{y}$ ). The final resultant vector can then be easily found from $R_{x}$ and $R_{y}$ .

Q: What happens to the components if the vector is in another quadrant? A: The signs of the components will change. For example, a vector in Quadrant II (angle between 90° and 180°) will have a negative x-component ( $A_{x} < 0$ ) and a positive y-component ( $A_{y} > 0$ ). The trigonometric formulas $A_{x} = A cos θ$ and $A_{y} = A sin θ$ automatically handle these signs as long as $θ$ is measured from the positive x-axis.

Q: Under what circumstances would a vector have equal magnitude components ( $A_{x} = A_{y}$ )? A: When the vector makes an angle of 45° (or 135°, 225°, 315°) with the positive x-axis. In such cases, $cos θ = sin θ$ , so $A_{x} = A cos θ = A sin θ = A_{y}$ . Graphically, this occurs when the vector bisects the angle between the x and y axes.

Q: If

A_{x} = - 2

and

A_{y} = - 2

, at what angle does the resultant vector make with the positive x-axis? A: Both components are negative, placing the vector in Quadrant III. The reference angle is

ϕ = tan^{- 1} (\frac{∣ - 2∣}{∣ - 2∣}) = tan^{- 1} (1) = 45°

. Since the vector is in Quadrant III, the actual angle is

180° + 45° = 225°

Summary

Vector Resolution is the process of splitting a vector into its perpendicular (rectangular) components.

The components are typically aligned with the x and y-axes.

The magnitudes of the components are found using trigonometry:

$A_{x} = A cos θ$
$A_{y} = A sin θ$

This technique simplifies complex vector problems, particularly vector addition, by allowing us to work with simple scalar quantities (the component magnitudes) along each axis independently.

The rectangular component method is extensively used in analyzing Equilibrium→ of forces.

Concept	Formula
x-component	$A_{x} = A cos θ$
y-component	$A_{y} = A sin θ$
Vector Magnitude	$A = A_{x}^{2} + A_{y}^{2}$
Vector Direction	$θ = tan^{- 1} (\frac{A _{y}}{A _{x}})$