Skip to main content
Ch 03: Vectors and Coordinate Systems
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 03: Vectors and Coordinate SystemsProblem 24a
Chapter 3, Problem 24a

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

Verified step by step guidance
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
Recommended video:
Guided course
06:44
Adding 3 Vectors in Unit Vector Notation

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|).
Recommended video:
Guided course
05:56
Calculating the Angle Between 2 Vectors Using the Dot Product

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.
Recommended video:
Guided course
05:17
Coordinates of Center of Mass of 4 objects
Related Practice
Textbook Question

The position of a particle as a function of time is given by r→\(\overrightarrow{r}\) = ( 5.0î ＋4.0ĵ )t² m where t is in seconds. What is the particle's speed at t = 0, 2, and 5 s?

2136
views
Textbook Question

Use geometry and trigonometry to determine the magnitude and direction of G = E＋F.

1808
views
Textbook Question

The position of a particle as a function of time is given by r→\(\overrightarrow{r}\) = ( 5.0î ＋4.0ĵ )t² m where t is in seconds. Find an expression for the particle's velocity v→\(\overrightarrow{v}\) as a function of time.

2820
views
Textbook Question

The position of a particle as a function of time is given by r→\(\overrightarrow{r}\) = ( 5.0î ＋4.0ĵ )t² m where t is in seconds. What is the particle's distance from the origin at t = 0, 2, and 5 s?

1943
views
Textbook Question

FIGURE P3.26 shows vectors A and B. Find D = 2A ＋B Write your answer in component form.

2142
views
Textbook Question

Use components to determine the magnitude and direction of G = E＋F.

1634
views