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• What are the position vectors for points A and B?

  rA = 2ˆi + αˆj + ˆk, rB = ˆi + 3ˆj − 8ˆk
• What do the unit vectors ˆi, ˆj, and ˆk represent?

  They point along the x, y, and z directions
• How do you calculate the dot product rA.rB?

  By multiplying corresponding components and summing
• What is the formula for the dot product of two vectors?

  rA.rB = A_xB_x + A_yB_y + A_zB_z
• What does it mean for vectors rA and rB to be perpendicular?

  Their dot product equals zero
• How can you determine the value of α for perpendicular vectors?

  Set rA.rB = 0 and solve for α
• What is the formula to find the angle that vector rB makes with the x-axis?

  θ = arccos(B_x / |rB|)
• How do you calculate the work done W by force F moving from A to B?

  W = F . (rB - rA)
• What is the formula for the moment M of a force about a vector?

  M = rB × F
• How do you find the magnitude of the vector M?

  Calculate |M| = |rB × F|
• What are the equations for the motion of a bead along a string?

  x(t) = sin(t), y(t) = cos(t)
• What is the trajectory described by x²(t) + y²(t)?

  A circle with radius 1
• How do you calculate the derivatives dx/dt and dy/dt for the bead's motion?

  Differentiate x(t) and y(t) with respect to t
• How do you show that dy/dx = -tan(t)?

  Use the chain rule on x(t) and y(t)
• What is the second derivative d²y/dx² in terms of y?

  d²y/dx² = -1/y³
• What is the formula for integration by parts?

  ∫uv' dx = uv - ∫u'v dx
• What is the integral of x e^x dx?

  x e^x - e^x + C
• How do you calculate the definite integral ∫₀¹ x² e^x dx?

  Use integration by parts twice
• What is the probability of a water pump showing a defect?

  1/10
• How do you find the probability of exactly 3 pumps showing a defect?

  Use the binomial probability formula
• How do you calculate the probability that exactly 11 pumps do not present a defect?

  Use the binomial probability formula
• How do you find the probability of less than 3 pumps showing a defect?

  Sum probabilities for 0, 1, and 2 defects