Resolution of Vectors

Vector Resolution is splitting a vector into its components along different coordinate axes. When a vector is expressed in terms of its components, it becomes easier to analyze its effects in different directions. This process is particularly useful when dealing with vector quantities such as forces, velocities etc.

Vector resolution is a powerful tool in physics and engineering, that enables the analysis and prediction of complex physical phenomena by simplifying vectors into manageable components.

What are Vectors?

In mathematics and physics, a vector is a mathematical object that has magnitude(or length) and direction and obeys the vector law of addition. Vectors are often represented as arrows in space with a particular length(as magnitude) and a particular direction while quantities that have both direction and magnitude are called scalar quantities.

Vectors can be used to represent various physical quantities such as displacement, velocity, acceleration and force.

What is Vector Resolution?

Vector Resolution refers to the process of breaking down a vector into its components along specified axes or directions. This decomposition allows for simplified calculations or analyzing the effects of a vector in different directions. This involves decomposing the vector into two perpendicular components, often along the horizontal and vertical axes.

For instance, if you have a vector representing a force acting at an angle to a surface, you might resolve it into horizontal and vertical components to understand how much of the force is acting in each direction. As we will see in the next section, this can be done using trigonometry. The concept of vector resolution is fundamental to both physics and engineering, especially when studying forces, velocities, and other vector quantities.

To successfully carry out vector resolution, we must understand vector addition. 2 different laws provide geometric interpretations of vector addition:

• Triangle Law of Vector Addition
• Parallelogram Law of Vector Addition

Triangle Law of Vector Addition

This law states that,

If two vectors are represented as two sides of a triangle, then their resultant (or sum) vector is represented by the third side of the triangle, taken from the tail of the first vector to the head of the second vector.

Parallelogram Law of Vector Addition

This law states that,

If two vectors are represented as two adjacent sides of a parallelogram, then their resultant (or sum) vector is represented by the diagonal of the parallelogram, taken from the common starting point of the two vectors to their opposite endpoint.

Rectangular Components of a Vector

Rectangular components are the projections of that vector onto the coordinate axes of a Cartesian coordinate system. In a two-dimensional Cartesian coordinate system, a vector can be decomposed into its horizontal and vertical components.

Horizontal Component

Horizontal component lies along the x-axis. If the angle between the vector and the horizontal axis is θ, then:

Horizontal Component = Magnitude of Vector × cos(θ)

Vertical Component

Vertical component lies along the y-axis. If the angle between the vector and the horizontal axis is θ, then:

Vertical Component = Magnitude of Vector × sin(θ)

Method of Resolving a Vector

Vectors can be easily resolved into its rectangular components and the steps for the same are,

Step 1: Identify the Axes: Start by identifying the axes along which you want to resolve. Typically, the axes are chosen to be perpendicular to each other, such as horizontal and vertical axes.

Step 2: Determine the Angle: Determine the angle it makes with the anti-clockwise direction of the axis.

Step 3: Use Trigonometry: If the angle between the vector and the horizontal axis is θ:

• Horizontal Component = Magnitude of the vector × cos(θ)
• Vertical Component = Magnitude of the vector × sin(θ)

Write down resolved vector components, specifying both their magnitudes and directions.

Trigonometric Method of Vector Resolution

Trigonometric method is a technique used to break down a vector into its components along specified directions using trigonometric functions. This method is particularly useful when dealing with vectors in two dimensions.

Let’s say we have a vector [Tex]\vec{v}[/Tex] with magnitude (v) that makes an angle θ with respect to positive x-axis. We want to find its rectangular components along x-axis (v_x) and y-axis (v_y).

• Horizontal Component (v_x): To find the horizontal component, we use the cosine of the angle θ

v_x = |v| × cos θ

• Vertical Component (v_y): To find the vertical component, we use the sine of the angle θ

v_y = |v| × sin θ

Applications of Vector Resolution

Application of Vector Resolution in Physics and Engineering are as,

Projectile Motion: When studying the motion of projectiles, such as objects thrown or launched into the air, vector resolution helps break down the initial velocity into horizontal and vertical components. This allows for analyzing the motion independently along each axis, making calculations more manageable.

Force Analysis: In mechanics, vector resolution is used to break down forces acting on an object into components along specified axes. This simplifies the analysis of forces, especially when dealing with forces acting at angles.

Statics and Dynamics: Vector resolution is essential in analyzing the equilibrium or motion of objects under the influence of multiple forces. By resolving forces into horizontal and vertical components, we can determine conditions for equilibrium or calculate the resulting motion.

Engineering applications of Vector Resolution are as follows:

Robotics: Vector resolution plays a vital role in robotics for analyzing the motion and forces acting on robotic manipulators and end-effectors. Engineers use it to decompose forces and velocities in robot kinematics and dynamics, enabling precise control and motion planning.

Fluid Mechanics: In fluid engineering applications, vector resolution is used to analyze fluid flow behavior, such as velocity profiles, pressure distributions, and shear forces. Engineers use it to decompose fluid velocities and forces into components, aiding in the design of pipelines, pumps, and hydraulic systems.

Common Mistakes of Resolution Vector

Common mistakes and misconceptions relating to Resolution of Vector are:

• While defining vectors, students usually miss out the vector law of addition.
• Steps outlined above will work successfully, and reduce the complexity of parallelogram or trigonometric methods.
• Students don’t cross-check their answer by adding the components.

Conclusion: Resolution Vector

Vector resolution involves breaking down vectors into components along different axes, typically horizontal and vertical. This simplifies analysis by isolating effects in specific directions. Trigonometric relationships aid in determining component magnitudes based on vector angles, which are then combined through addition to reconstruct the original vector.

Examples for Vector Resolution

Example 1: A body is moving with a velocity of 10 m/s at an angle of 60° to the horizontal. Break the velocity down into its horizontal and vertical components.

Solution:

Let’s horizontal component be V_x and vertical component be V_y .

Horizontal Component V_x

V_x = V_x cos θ

⇒ V_x = 10 x cos(60°) = 10 x 0.5

⇒ V_x = 5 m/s

Vertical Component V_y

V_y = V × sin θ

⇒ V_y = 10 × sin(60°) = 10 x √3/2

⇒ V_y = 5√3 m/s

So, horizontal component of velocity is 5 m/s and vertical component is 5√3 m/s.

Magnitude of Vector = √[(5)² + (5√3)²] = √(25 + 75) = 10 m/s

Angle = tan^-1(5√3/5) = 60°

Example 2: A force of 50N is acting on a body at an angle of 30 degrees with the horizontal What force is pushing the body forward?

Solution:

Let’s horizontal component of force be F_x

Horizontal Component F_x

F_x = F_x cos θ = 50 × cos(30°)

⇒ F_x = 50×√3/2

⇒ F_x = 25√3 N

So, force pushing the body forward is 25√3 N.

Practice Problems on Resolution of Vector

Various practice problems on resolution of vector are,

Problem 1: A force of 80 N is applied at an angle of 45 degrees with the horizontal. Determine the horizontal and vertical components of the force.

Problem 2: A velocity of 15 m/s is directed at an angle of 60 degrees with the horizontal axis. Find the horizontal and vertical components of the velocity.

Problem 3: An airplane is flying with a velocity of 300 km/h at an angle of 30 degrees with the horizontal. Determine the horizontal and vertical components of the velocity.

FAQs on Resolution Vector

How is Vector Resolution Used in Real Life?

In real life, vector addition is used in the fields of navigation, construction, aviation, weather forecasting and robotics.

Can Vector Resolution Be Applied to Non-Physical Fields?

Yes, vector resolution can be applied to non-physical fields as well. While vector resolution is commonly associated with physics and engineering, its principles can be extended to other disciplines where quantities have both magnitude and direction such as economics, echocardiography in medicine, computer graphics, machine learning, and signal processing.

What Are the key challenges in learning Vector Resolution?

The key challenges in learning vector resolution are understanding vector concepts, visualizing vectors in multiple dimensions, trigonometry and problem-solving skills and the transition from 2D to 3D spaces.

How Does Vector Resolution Differ in Two-Dimensional and Three-Dimensional Spaces?

In 2D space, vectors are resolved into two components: x-component and y-component. In 3D space, vectors are resolved into three components: x, y and z-components. In 2D we require only one angle, whereas in 3D we require two angles.