Vector Operations and Calculus

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Questions and Answers

What is the scalar projection of a onto b?

  • The magnitude of **b**
  • The dot product of **a** and **b**
  • The magnitude of **a**
  • The magnitude of the projection vector (correct)

What property of vector addition is demonstrated by the equation (a + b) + c = a + (b + c)?

  • Distributive
  • Commutative
  • Associative (correct)
  • Scalar multiplication

What is the result of the dot product a · (b + c)?

  • **a** · **b** + **b** · **c**
  • **b** · **c**
  • **a** · **b** + **a** · **c** (correct)
  • **a** · **b** - **a** · **c**

What is the gradient of a function f(x, y, z)?

<p>A vector pointing in the direction of the maximum rate of increase of <strong>f</strong> (A)</p> Signup and view all the answers

What does the divergence of a vector field measure?

<p>How much a vector field spreads out or converges at a point (C)</p> Signup and view all the answers

What is the definition of a unit vector?

<p>A vector with a magnitude of 1 (A)</p> Signup and view all the answers

What is the notation typically used for unit vectors?

<p>A hat symbol, e.g., <strong>î</strong> (D)</p> Signup and view all the answers

What is the property of a unit vector u that states u × u = 0?

<p>The cross product of <strong>u</strong> with itself is 0 (C)</p> Signup and view all the answers

What is the standard basis vector in the x-direction?

<p><strong>î</strong> (A)</p> Signup and view all the answers

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Study Notes

Vector Projections

  • Projection of a vector onto another vector:
    • Given vectors a and b, the projection of a onto b is defined as:
      • proj_b(a) = (a · b / ||b||^2) * b
    • This gives the component of a in the direction of b
  • Scalar projection:
    • The scalar projection of a onto b is the magnitude of the projection vector
    • Calculated as: a · b / ||b||

Vector Operations

  • Vector addition:
    • Commutative: a + b = b + a
    • Associative: (a + b) + c = a + (b + c)
  • Scalar multiplication:
    • Distributive: k(a + b) = ka + kb
    • Associative: (km)a = k(ma)
  • Dot product (inner product):
    • a · b = b · a (commutative)
    • a · (b + c) = a · b + a · c (distributive)

Vector Calculus

  • Gradient:
    • A vector pointing in the direction of the maximum rate of increase of a function
    • Calculated as: ∇f(x, y, z) = (∂f/∂x, ∂f/∂y, ∂f/∂z)
  • Divergence:
    • Measures how much a vector field spreads out or converges at a point
    • Calculated as: ∇ · F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z
  • Curl:
    • Measures the rotational tendency of a vector field around a point
    • Calculated as: ∇ × F = (∂F_z/∂y - ∂F_y/∂z, ∂F_x/∂z - ∂F_z/∂x, ∂F_y/∂x - ∂F_x/∂y)

Unit Vectors

  • Definition: A vector with a magnitude of 1
  • Notation: Typically denoted with a hat symbol, e.g., î, ĵ, Ä·
  • Properties:
    • ||u|| = 1
    • u · u = 1
    • u × u = 0
  • Examples:
    • î = <1, 0, 0> (standard basis vector in the x-direction)
    • ĵ = <0, 1, 0> (standard basis vector in the y-direction)
    • Ä· = <0, 0, 1> (standard basis vector in the z-direction)

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