Physics 20: 2.3 2D Vectors

Summary

This educational video script explains the process of solving vector addition in two dimensions with vectors at angles. The presenter guides viewers through the steps of determining vector components, combining them, and calculating the resultant vector. Using an example of a person walking 15 meters at 30° north of east and then 10 meters at 20° west of north, the script demonstrates how to find the horizontal and vertical components using trigonometry, add them to find the resultant displacement, and then use the Pythagorean theorem to determine the magnitude and direction of the final vector. The method is illustrated with clear examples, making it accessible for learners to understand and apply.

Takeaways

• 📐 Today, we are solving vector problems involving two vectors in two dimensions.
• 📊 First, break down the vectors into components to simplify the problem.
• 📝 Example problem: A person walks 15m at East 30° North and then 10m at North 20° West.
• 📏 Calculate the vertical and horizontal components of the first vector (15m at 30°).
• 📐 The vertical component is 7.5m and the horizontal component is 12.99m.
• 📏 Next, calculate the components of the second vector (10m at 20°).
• 📐 The vertical component is 9.4m and the horizontal component is 3.42m.
• 🔄 Combine the vertical components (7.5m + 9.4m = 16.9m) and the horizontal components (12.99m - 3.42m = 9.57m).
• 📐 Use the Pythagorean theorem to find the resultant vector length (19m) and angle (60°).
• ✅ Final displacement is 19m at East 60° North.

Q & A

What is the process for solving vector addition problems with two vectors in two dimensions?

The process involves breaking down the problem into steps: first, determine the vector components for each vector, then combine these components, and finally calculate the resultant vector which includes its magnitude and direction.

How does the example in the script illustrate the vector addition process?

The example demonstrates the process by showing a person walking 15 meters at 30° North of East, followed by 10 meters at 20° West of North. The script then guides through finding the horizontal and vertical components of each vector, combining them, and finally calculating the resultant vector's magnitude and direction.

What is the first step in solving for the vector components of a given vector?

The first step is to sketch the situation and determine the horizontal (x) and vertical (y) components of the vector by using trigonometric functions such as sine for the vertical component and cosine for the horizontal component.

How are the horizontal and vertical components of the first vector in the example calculated?

For the first vector, the vertical component is calculated using sin(30°) * 15 m, which equals 7.5 m. The horizontal component is calculated using cos(30°) * 15 m, which equals approximately 12.99 m.

What trigonometric functions are used to find the components of the second vector in the example?

For the second vector, sine is used to find the horizontal component (sin(20°) * 10 m), and cosine is used to find the vertical component (cos(20°) * 10 m).

How are the components of the second vector combined with the first to find the resultant vector?

The vertical components (9.4 m and 7.5 m) are added together, and the horizontal components (12.99 m to the right and 3.42 m to the left) are combined by subtracting the leftward component from the rightward component, resulting in a net horizontal component of 9.57 m to the right.

What mathematical theorem is used to find the magnitude of the resultant vector?

The Pythagorean theorem is used to find the magnitude of the resultant vector by adding the squares of the combined vertical and horizontal components and then taking the square root of the sum.

How is the direction of the resultant vector determined?

The direction of the resultant vector is determined by calculating the angle it makes with the horizontal axis, which can be found using trigonometric functions and the components of the resultant vector.

What is the final answer for the magnitude and direction of the resultant vector in the example?

The final answer for the magnitude is approximately 19.4 meters, and the direction is East 60° North.

Why is it important to break down the original vectors into their components before combining them?

Breaking down the original vectors into their components simplifies the process of vector addition, especially when the vectors are not aligned with the coordinate axes. It allows for easier calculation of the resultant vector's magnitude and direction.

Can the process demonstrated in the script be applied to vectors with different magnitudes or angles?

Yes, the process can be applied to vectors with different magnitudes or angles. The key is to calculate the horizontal and vertical components for each vector and then combine and recalculate the resultant vector accordingly.