To add vectors that are at a right angle to each other, you can use the Pythagorean theorem to find the magnitude of the resultant vector and trigonometric functions to find its direction.
Understanding Vector Addition at Right Angles
When two vectors are perpendicular, they form a right triangle with the resultant vector as the hypotenuse. Here's a detailed approach:
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Visual Representation:
- Imagine two vectors, A and B, starting from the same point and forming a 90-degree angle.
- The resultant vector, R, extends from the starting point to the endpoint of the second vector, forming the hypotenuse of the right triangle.
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Finding the Magnitude (Length) of the Resultant Vector:
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Use the Pythagorean theorem: R² = A² + B²
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Therefore, the magnitude of the resultant vector, R, is: R = √(A² + B²)
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Example: According to the provided reference, if one vector is 30 and the other is 1.25, the resultant vector's magnitude would be calculated as the square root of 30 + 1.25, which results in approximately 5.6.
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Determining the Direction of the Resultant Vector:
- Use trigonometric functions (sine, cosine, or tangent). The choice depends on the information available.
- Usually, you'll use the tangent function to find the angle (θ) the resultant vector makes with one of the original vectors: tan(θ) = Opposite / Adjacent
- In our example, let's assume we want the angle relative to the vector represented by 'A'. So, tan(θ) = B / A.
- Then θ = tan⁻¹(B / A).
Steps for Vector Addition
Here’s a step-by-step approach, along with practical insights:
- Step 1: Draw a diagram of the two vectors starting from the same point, forming a right angle.
- Step 2: Calculate the magnitude of the resultant vector R using the Pythagorean theorem: R = √(A² + B²).
- Step 3: Determine the direction (angle θ) of the resultant vector using tangent, where tan(θ) = B/A, so θ = tan⁻¹(B / A)
Practical Implications
- Navigation: Understanding how to add vectors at right angles is crucial in navigation, for calculating the overall direction and distance travelled when moving in two directions, one perpendicular to the other.
- Physics: Vector addition is a foundational concept in physics to understand motion, forces, and fields.
- Engineering: It's used in designing structures, machines, and many other complex systems where forces act at angles.
By following these steps, one can effectively add vectors that are at right angles to each other. Remember to always express the magnitude and direction to fully describe the resultant vector.
