Physics Vectors Problem Set

Study Notes

Problem 2: Resultant Vector

The magnitude of the resultant vector (R) is equal to F₂, which is 10 N.
The direction of R makes a 45° angle with F₁.
The angle between F₂ and R is 45°.
The magnitude of F₁ can be calculated using the law of cosines: F₁² = R² + F₂² - 2(R)(F₂)cos(45°) = 10² + 10² - 2(10)(10)cos(45°) = 100 + 100 - 200(√2/2) = 100(2 - √2)
Therefore, the magnitude of F₁ is √(100(2 - √2)) N.

Problem 3: Vectors with Equal Magnitude

Two vectors with equal magnitude, y, have a ratio of their sum to difference equal to √3.
This ratio implies the angle (θ) between the vectors is related to the tangent of half the angle: tan(θ/2) = 1/√3.
Solving for θ, the angle between the vectors is 60°.

Problem 4: Resultant of Four Force Vectors

Four force vectors, F₁, F₂, F₃, and F₄ are acting at a single point.
F₁ acts horizontally along the positive X-axis with a magnitude of 10 N.
F₂ acts vertically downwards with a magnitude of 7 N.
F₃ acts horizontally along the positive X-axis with a magnitude of 2 N, at an angle of 57° to F₁.
F₄ acts vertically downwards with a magnitude of 15 N, at an angle of 37° to F₁.
To determine the resultant vector, you need to resolve each vector into its horizontal and vertical components and sum them individually.
The resultant vector can then be calculated using the Pythagorean theorem and its direction using the arctangent function.

Problem 5: Vector Operations

Two vectors are given: a = 4i - 3j + 2k and b = -3i + 2j - 2k.
The dot product (a + b) ⋅ (a × 3b) needs to be calculated.
First, calculate the sum of the two vectors: a + b = (4 - 3)i + (-3 + 2)j + (2 - 2)k = i - j
Then calculate the cross product of a and 3b: a × 3b = (4i - 3j + 2k) × 3(-3i + 2j - 2k) = (-12i - 24j - 6k)
Finally, calculate the dot product of (a + b) and (a × 3b): (i - j) ⋅ (-12i - 24j - 6k) = -12 + 24 = 12
Therefore, the result of (a + b) ⋅ (a × 3b) is 12.

Description

This quiz focuses on solving problems related to vectors, including resultant vectors and the law of cosines. It includes scenarios involving angles, force magnitudes, and vector relationships. Perfect for physics students looking to enhance their understanding of vector mechanics.