How to Resolve Vectors in Physics

Vectors are used extensively in physics to describe the magnitude and direction of quantities such as displacement, velocity, and force. To solve problems involving vectors, it is often necessary to resolve them into their components. In this article, we will discuss how to resolve vectors in physics and provide examples to help HSC physics students understand this concept.

What is Vector Resolution?

Vector resolution is the process of breaking down a vector into its components along two or more axes. These components can then be analysed separately using the laws of vector addition and subtraction. Vector resolution is important in physics because it allows us to analyse the motion of an object in two or more dimensions.

Resolving Vectors into Components

To resolve a vector into its components, we need to know the magnitude and direction of the vector, as well as the angles between the vector and the axes of the coordinate system (e.g. north, east, south and west).

When resolving a vector on a two-dimensional plane, the vector to be resolved is to become the hypotenuse of a right angled triangle, and the components become the two perpendicular sides.

Let's consider an example to illustrate this concept:

Example 1: An object is fired at 10 `m s^{-1}` at 40º above the horizontal. Resolve the force into its horizontal and vertical components.

Here's how to do it:

Step 1: Draw a diagram

First, we need to draw a diagram of the vector and the axes of the coordinate system, as shown below:

Step 2: Determine the magnitudes of the components

Next, we need to determine the magnitudes of the horizontal and vertical components using trigonometry. The horizontal component can be found using the cosine function, while the vertical component can be found using the sine function:

Horizontal component = 10 cos θ = 10 cos 40° = 7.66 `m s^{-1}`

Vertical component = 10 sin θ = 10 sin 40° = 6.43 `m s^{-1}`

Therefore, the horizontal component is 7.66 m/s to the right, and the vertical component is 6.43 m/s upward.

Finding Resultant Vector

Once we have resolved vectors into their components, we can use the laws of vector addition to find the resultant vector. The resultant vector is the vector that represents the sum of the individual vectors.

The resultant vector is formed by drawing an arrow from the tail of one component to the head of the other.

The magnitude of the resultant vector can be calculated using Pythagoras' theorem. The direction of the resultant vector can be calculated using trigonometry.

Example 2: A boat is traveling at a speed of 10 m/s due north. There is a current flowing at a speed of 5 m/s due east. What is the boat's resultant velocity?

Solution:

The resultant velocity considers the velocity of the boat as well as the current. To find the resultant vector from vector addition, you need to follow these steps:

Step 1: Draw a diagram

Draw a diagram of the vectors you want to add, making sure to label the magnitude and direction of each vector. You can use a scale to ensure the lengths of the vectors are proportional to their magnitudes.

The resultant vector is drawn such that is becomes the hypotenuse of a right angled triangle.

Step 2: Calculate the magnitude of the resultant vector

Use the Pythagorean theorem to find the magnitude of the resultant vector.

$$v^2 = 5^2 + 10^2$$

$$v = \sqrt{5^2+10^2}$$

$$v = 11.2 m s^{-1}$$

 

Step 3: Calculate the direction of the resultant vector.

Since velocity is a vector quantity, the direction is required in addition to the magnitude. The direction of the resultant vector can be found using trigonometry.

The angle between the resultant vector and the x-axis (east-west) is given by:

$$\tan{\theta}=\frac{10}{5}$$

$$\theta = 85º$$

 

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