Physics Vectors Problem Set

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.