What are the formulas related to vectors, including unit vectors, vector addition, and vector products?
Understand the Problem
The image contains various vector formulas, including unit vector, triangle and parallelogram law of vector addition, resolution of vector in two and three dimensions, dot product, and cross product.
Answer
The image shows formulas for unit vectors, vector addition laws, resolution of vectors in 2D and 3D, dot product, and cross product.
The image shows the following vector formulas:
- Unit Vector: Â = A / |A|
- Triangle and Parallelogram law of vector addition: Magnitude R = √(A² + B² + 2ABcosθ), Direction α = tan⁻¹ (B sin θ / A+B cos θ)
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Resolution of Vector:
- In two dimensions: A = √(Aₓ² + Aᵧ²), Direction θ = tan⁻¹(Aᵧ / Aₓ)
- In three dimensions: A = √(Aₓ² + Aᵧ² + A_z²)
- Dot Product or Scalar Product: A·B = |A| |B| cos θ
- Cross or Vector Product: A × B = |A| |B| sin θ n̂
Answer for screen readers
The image shows the following vector formulas:
- Unit Vector: Â = A / |A|
- Triangle and Parallelogram law of vector addition: Magnitude R = √(A² + B² + 2ABcosθ), Direction α = tan⁻¹ (B sin θ / A+B cos θ)
-
Resolution of Vector:
- In two dimensions: A = √(Aₓ² + Aᵧ²), Direction θ = tan⁻¹(Aᵧ / Aₓ)
- In three dimensions: A = √(Aₓ² + Aᵧ² + A_z²)
- Dot Product or Scalar Product: A·B = |A| |B| cos θ
- Cross or Vector Product: A × B = |A| |B| sin θ n̂
More Information
These formulas cover the basic operations with vectors, which are foundational in vector algebra and are widely used in fields like physics and engineering.
Tips
Common mistakes include not normalizing vectors correctly for unit vectors or using incorrect angles in dot and cross product calculations.
Sources
- 8.8: Vectors - Mathematics LibreTexts - math.libretexts.org
- Vector Formulas - Superprof - superprof.co.uk
- Formula, Vector Sum | Addition of Vectors - Cuemath - cuemath.com
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