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Ch 03: Vectors and Coordinate Systems
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 03: Vectors and Coordinate SystemsProblema 24a
Capitolo 3, Problema 24a

What is the angle Φ between vectors E and F in FIGURE P3.24?

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1
Step 1: Identify the components of vectors E and F from the graph. Vector E (red) has components E_x = 1 and E_y = 3, while vector F (blue) has components F_x = -1 and F_y = 1.
Step 2: Use the dot product formula to calculate the dot product of vectors E and F. The formula is E • F = E_x * F_x + E_y * F_y.
Step 3: Calculate the magnitudes of vectors E and F using the formula |E| = sqrt(E_x^2 + E_y^2) and |F| = sqrt(F_x^2 + F_y^2).
Step 4: Use the formula for the angle between two vectors: cos(Φ) = (E • F) / (|E| * |F|). Substitute the values obtained from Steps 2 and 3 into this formula.
Step 5: Solve for Φ by taking the inverse cosine (arccos) of the result from Step 4. This will give the angle between vectors E and F.

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Vector Representation

Vectors are quantities that have both magnitude and direction, represented graphically as arrows. The length of the arrow indicates the vector's magnitude, while the arrowhead shows its direction. In the context of the question, vectors E and F are represented in a coordinate system, allowing for the analysis of their relationship through the angle between them.
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Angle Between Vectors

The angle between two vectors can be determined using the dot product formula, which relates the cosine of the angle to the magnitudes of the vectors and their dot product. This angle, denoted as Φ, is crucial for understanding how the vectors interact, such as in physics applications involving forces or velocities. The angle can be calculated using the formula: cos(Φ) = (A · B) / (|A| |B|).
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Coordinate System

A coordinate system provides a framework for locating points in space using numerical values. In this case, the Cartesian coordinate system is used, where each point is defined by its x and y coordinates. Understanding this system is essential for accurately plotting vectors and determining the angle between them, as it allows for visual representation and mathematical calculations.
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