- #1
brinlin
- 12
- 0
Components, Projection, and Resolution
- Context: MHB
- Thread starter brinlin
- Start date
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SUMMARY
The discussion focuses on understanding vector components, specifically the "component of u along v" using trigonometric principles. It emphasizes the geometric interpretation of vectors u and v, where the component is determined by the length of the adjacent side of a right triangle formed by the vectors. The dot product formula, $u \cdot v = |u||v| \cos(\theta)$, is highlighted as a key method for calculating this component. Mastery of these concepts is essential for effectively applying vector analysis in various mathematical and engineering contexts.
PREREQUISITES- Understanding of basic trigonometry, including sine and cosine functions.
- Familiarity with vector notation and operations, specifically dot products.
- Knowledge of geometric interpretations of vectors in two or three dimensions.
- Ability to visualize right triangles and their properties in relation to vectors.
- Study the geometric interpretation of vector addition and subtraction.
- Learn about the properties and applications of the dot product in physics and engineering.
- Explore the concept of vector projections and their significance in various fields.
- Investigate advanced trigonometric identities and their applications in vector analysis.
Students in mathematics, physics, and engineering, as well as professionals working with vector analysis and trigonometric applications.
- #2
HOI
- 921
- 2
Where are you getting these problems? You seem to be saying you know nothing at all about trigonometry!
Imagine the vectors u and v with their "tails" together. Draw a perpendicular from the tip of u to v. The "component of u along v" is the distance from the common tails to that perpendicular. So you have a right triangle where the length of the hypotenuse is the length of u and one angle is the angle between u and v. The "component of u along v" is the length of the "near side" of that triangle so you will need the cosine of the two vectors. If you know that the dot product of two vectors is $u\cdot v= |u||v| cos(\theta)$ then that will be easy to calculate.
Imagine the vectors u and v with their "tails" together. Draw a perpendicular from the tip of u to v. The "component of u along v" is the distance from the common tails to that perpendicular. So you have a right triangle where the length of the hypotenuse is the length of u and one angle is the angle between u and v. The "component of u along v" is the length of the "near side" of that triangle so you will need the cosine of the two vectors. If you know that the dot product of two vectors is $u\cdot v= |u||v| cos(\theta)$ then that will be easy to calculate.
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