Adding and Subtracting Coplanar Vectors
Vectors represent quantities characterised by both magnitude and direction. The operations of vector addition and subtraction are foundational in physics for combining or comparing these quantities.
Addition of Vectors
- Tip-to-Tail Method:
- Positioning: Align the second vector's tail to the tip of the first vector.
- Drawing the Resultant: The resultant vector is drawn from the tail of the first vector to the tip of the second vector.
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- Magnitude and Direction: Calculate the magnitude using the Pythagorean theorem if the vectors are perpendicular. The direction is determined using trigonometry.
- Example: Consider Vector A (3 units East) and Vector B (4 units North). By positioning B at the tip of A, the resultant vector, Vector R, is formed, extending diagonally, indicating a north-east direction.
Subtraction of Vectors
- Reversing and Adding:
- Reverse the vector you are subtracting.
- Use the tip-to-tail method to add this reversed vector to the first vector.
- The resultant vector's direction and magnitude are determined similarly to vector addition.
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- Example: Subtracting Vector B (3 units North) from Vector A (5 units East). Reverse B to point 3 units South, then add it to A using the tip-to-tail method, resulting in a south-east directed vector.
Resolving Vectors into Components
Breaking down vectors into their perpendicular components (usually horizontal and vertical) simplifies many physics problems.
Horizontal and Vertical Components
- Methodology:
- Draw the vector at the appropriate angle.
- From the tail, draw a horizontal line (x-component).
- From the tip, draw a vertical line to intersect the horizontal (y-component).
- The lengths of these lines represent the components of the vector.
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- Calculations: Use sine for the vertical component and cosine for the horizontal component, with the vector's magnitude and angle. For instance, for a vector of magnitude V at an angle θ, the horizontal component is V cos(θ), and the vertical component is V sin(θ).
Practical Exercises
Engaging in practical exercises fortifies the understanding of vector operations.
Exercise 1: Vector Addition
- Problem: Add Vector A (4 units, 30° North of East) and Vector B (5 units, 45° West of North).
- Solution:
- Resolve Components: For A, the x-component is 4 cos(30°), and the y-component is 4 sin(30°). For B, calculate similarly.
- Add Components: Add the corresponding x and y components of A and B.
- Resultant Vector: Use Pythagorean theorem for magnitude and arctan for direction.
Exercise 2: Vector Subtraction
- Problem: Subtract Vector C (6 units, East) from Vector D (8 units, 60° North of East).
- Solution:
- Resolve Vector C into components and reverse its direction.
- Add this reversed Vector C to Vector D's components.
- Calculate the resultant vector using the Pythagorean theorem and arctan for direction.
FAQ
Yes, vectors can be added in any order due to the commutative property of vector addition. This property states that the sum of two or more vectors remains unchanged regardless of the order in which they are added. For example, if you have three vectors A, B, and C, adding them in the order A + B + C will yield the same resultant vector as adding them in any other order, like B + C + A or C + A + B. This is because vectors are not bound to a fixed position in space; they can be moved parallel to themselves without changing their magnitude or direction, allowing for flexibility in the order of addition.
Resolving vectors into components is a fundamental technique in vector operations because it simplifies the addition, subtraction, and other manipulations of vectors. When a vector is broken down into its perpendicular components (typically horizontal and vertical), it becomes easier to work with, especially in calculations. For instance, in problems involving forces at angles, resolving a force into its horizontal and vertical components allows for the application of Newton's laws in each direction independently. This process is crucial in solving complex problems where direct vector addition or subtraction is not feasible or where vectors need to be analysed in terms of their individual components.
Vector subtraction is particularly applicable in situations where you need to determine the difference in quantities or the net effect of opposing quantities. A common example is when calculating the net force acting on an object in situations where forces are acting in opposite directions. Another instance is in displacement problems, where you might need to find the displacement from an initial to a final position, effectively subtracting one displacement vector from another. Vector subtraction is also used in fields such as electromagnetism and fluid dynamics, where it is important to understand the relative directions and magnitudes of vector quantities like electric fields or velocities.
For vectors that are not perpendicular, you cannot directly apply the Pythagorean theorem to find the magnitude of the resultant vector. Instead, you must first resolve each vector into its horizontal and vertical components. This is done using trigonometric functions, sine, and cosine, based on the angle each vector makes with a reference direction (usually the horizontal axis). After resolving the vectors, add or subtract their corresponding components. The resultant vector is then determined by finding the magnitude of the combined components using the Pythagorean theorem and the direction using the arctan function.
When adding more than two vectors, the direction of the resultant vector is determined by the overall direction after all the vectors have been added using the tip-to-tail method. Firstly, each vector is placed such that the tail of the next vector is at the tip of the previous one. After all vectors are connected in sequence, the resultant vector is drawn from the tail of the first vector to the tip of the last. To find the direction of this resultant vector, you may need to use trigonometry. This involves breaking down the resultant vector into its x and y components, calculating these components for each vector, summing them up, and then using arctan to find the angle of the resultant vector from the horizontal.
Practice Questions
The magnitude of the resultant displacement can be found using the Pythagorean theorem, as the two displacements are perpendicular to each other. The magnitude (R) is √(300² + 400²) = √(90000 + 160000) = √250000 = 500 km. To find the direction, the angle (θ) north of east can be calculated using tan θ = opposite/adjacent = 300/400. Therefore, θ = arctan(300/400) = 36.9°. Thus, the plane's resultant displacement is 500 km at 36.9° north of east.
To find A - B, first reverse B to point due east and then add it to A. Resolve A into components: Ax = 5 cos(30°) = 4.33 units (east), Ay = 5 sin(30°) = 2.5 units (north). Vector B, now reversed, has components B_x = 7 units (east) and By = 0. Adding A and B gives Rx = Ax + Bx = 4.33 + 7 = 11.33 units, and Ry = Ay + By = 2.5 + 0 = 2.5 units. The magnitude of R = √(11.33² + 2.5²) ≈ 11.63 units. The direction θ = arctan(2.5/11.33) ≈ 12.3° north of east. So, the magnitude of A - B is approximately 11.63 units at an angle of 12.3° north of east.
