Vectors, Motion, Complex Numbers, Probability

• What is a unit vector?

  A vector with a length of 1
• What is the purpose of the dot product?

  To measure how much two vectors point together
• What does a unit vector indicate?

  It indicates a specific direction
• What is vector notation?

  The method used to express vectors
• How does vector notation differ from unit vector notation?

  Vector notation expresses components, unit vectors indicate direction
• What is the equation of the green line in the graph?

  The equation of the green line is:

  y = 0.25x + 0.00
• If the dot product of vectors $r_{A}$ and $r_{B}$ is zero, what can be concluded?


  The vectors are orthogonal
• What are the unit vectors ˆi and ˆj in component form?

  ˆi = (1, 0), ˆj = (0, 1)
• How is a unit vector typically denoted?

  With a hat over the vector
• How would you express the vector (2, -3) in unit vector notation?

  As 2ˆi - 3ˆj
• How do you calculate $\cos \theta$ for vectors $\vec{a} =$$(1,2)$ and $\vec{b} =$$(3,1)$?


  $\cos \theta =$$\frac{5}{\sqrt{50}}$
• What does it mean for two vectors to be orthogonal?

  They are perpendicular to each other
• What angle do orthogonal vectors form?

  90° angle
• What is the value of the y-coordinate at the point marked with the blue dot?

  0.00
• What is the formula to calculate the cosine of the angle between two vectors?

  $\cos \theta =$$\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$
• What happens to the work done if the car doesn't move much?

  You do less work if it doesn't move much
• What is the value of the x-coordinate at the point marked with the blue dot?

  0.25
• How can the magnitude of the moment Mo be calculated based on the information given in the image?

  The magnitude of the moment Mo can be calculated as:

  Mo = F * r * sin(θ)

  where F is the force, r is the distance from the axis of the moment to the line of action of the force, and θ is the angle between the force and the line of action.
• How is the direction of the moment determined?

  By using the right-hand rule
• What are the definitions and notations for unit vectors and vector notation?

  • Unit Vector:
  • Definition: A vector with a magnitude of 1 indicating direction.
  • Notation: Denoted by a hat (e.g., ˆi, ˆj, ˆk).
  • Vector Notation:
  • Definition: Method to express vectors in components.
  • Notation: Coordinate pairs (e.g., (2, -3)) or unit vectors (e.g., 2ˆi - 3ˆj).
• What is the condition for two vectors to be orthogonal?

  Their dot product equals zero
• How do the components of Car A and Car B contribute to the dot product calculation?

  • Multiply east components: \(3 \times 5 = 15\)
  • Multiply north components: \(4 \times 2 = 8\)
  • Add results: \(15 + 8 = 23\)
• What is the formula to find the angle between two vectors using the dot product?

  $\cos \theta =$$\frac{\vec{a} \cdot \vec{b}}{| \vec{a} | | \vec{b} |}$
• If Car A goes 3 blocks east and 4 blocks north, what are its components?

  (3, 4)
• What is the formula to calculate the sine of the angle between two vectors?

  $\sin \theta =$$\frac{|\mathbf{a} \times \mathbf{b}|}{|\mathbf{a}||\mathbf{b}|}$
• Given vectors $\vec{a} =$$(1,2)$ and $\vec{b} =$$(3,1)$, what is the dot product?


  5
• What is the formula to find the angle between two vectors using the cross product?

  $\sin \theta =$$\frac{|\vec{a} \times \vec{b}|}{| \vec{a} | | \vec{b} |}$
• What do we integrate with respect to in parametric equations?

  Parameter t
• What is the equation of the red line in the graph?

  The equation of the red line is:

  y = -0.75x - 1.00
• How can you use the cosine and sine formulas to find the angle between two vectors?

  1. Calculate the dot product a·b and the magnitudes |a| and |b|
  2. Plug these into the cosine formula to find cos(θ)
  3. Plug the vectors a and b into the cross product formula to find |a×b|
  4. Plug this into the sine formula to find sin(θ)
  5. Use the inverse trigonometric functions to find the angle θ from the cos(θ) and sin(θ) values
• How can you find the area of the region R enclosed by the loop of the curve C?

  1. Use integration to find the area:

  $\int_{-\sqrt{9}}^{\sqrt{9}} (9 - t^2) dt$

  1. Evaluate the integral:

  $\int_{-3}^{3} (9 - t^2) dt =$$90t - \frac{t^3}{3}|_{-3}^{3} =$$90(3) - \frac{3^3}{3} - (-90(-3) - \frac{(-3)^3}{3}) =$$270 - 27 - (-270 - 27) =$$540$
• How can you find the x-coordinate at the points A and B where the curve C cuts the x-axis?

  1. Set y = 0 in the equation of the curve C: x = 5t^2 - 4, y = (9 - t^2)
  2. Substitute y = 0 to get: 0 = 9 - t^2
  3. Solve for t: t = ±3
  4. Substitute t = ±3 into the x-equation to get the x-coordinates:

  • At point A: x = 5(3)^2 - 4 = 41
  • At point B: x = 5(-3)^2 - 4 = 41
• What are the key variables in the formulas for the angle between two vectors?

  • a, b: The two vectors
  • |a|, |b|: The magnitudes (lengths) of the vectors
  • a·b: The dot product of the vectors
  • |a×b|: The magnitude of the cross product of the vectors
  • θ: The angle between the two vectors
• What is the significance of the dot product condition in physics and mathematics?

  • Determines if vectors are at right angles
  • Important for analyzing forces, velocities, and positions
• What happens if you give the pen different instructions for $x$ and $y$?


  You get crazy shapes like squiggles
• What is the equation of the curve C shown in the figure?

  x = 5t^2 - 4, y = (9 - t^2)
• What does a constant second derivative of 6 imply about $f(x)$?


  The rate of change increases at a constant rate
• What is the formula for the dot product of vectors rA and rB?

  rA · rB = (A1 × B1) + (A2 × B2)
• What are the two methods to calculate the angle between vectors?

  • Dot product formula
  • Cross product formula
• What does a dot product of 23 indicate about the movement of Car A and Car B?

  • The cars are moving in similar directions
  • They make progress of 23 grid blocks together