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Questions and Answers
What is the scalar projection of a onto b?
What is the scalar projection of a onto b?
What property of vector addition is demonstrated by the equation (a + b) + c = a + (b + c)?
What property of vector addition is demonstrated by the equation (a + b) + c = a + (b + c)?
What is the result of the dot product a · (b + c)?
What is the result of the dot product a · (b + c)?
What is the gradient of a function f(x, y, z)?
What is the gradient of a function f(x, y, z)?
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What does the divergence of a vector field measure?
What does the divergence of a vector field measure?
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What is the definition of a unit vector?
What is the definition of a unit vector?
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What is the notation typically used for unit vectors?
What is the notation typically used for unit vectors?
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What is the property of a unit vector u that states u × u = 0?
What is the property of a unit vector u that states u × u = 0?
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What is the standard basis vector in the x-direction?
What is the standard basis vector in the x-direction?
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Study Notes
Vector Projections
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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
- Given vectors a and b, the projection of a onto b is defined as:
-
Scalar projection:
- The scalar projection of a onto b is the magnitude of the projection vector
- Calculated as: a · b / ||b||
Vector Operations
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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
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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, ķ
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Properties:
- ||u|| = 1
- u · u = 1
- u × u = 0
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Examples:
- î = <1, 0, 0> (standard basis vector in the x-direction)
- ĵ = <0, 1, 0> (standard basis vector in the y-direction)
- ķ = <0, 0, 1> (standard basis vector in the z-direction)
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Description
Test your understanding of vector projections, operations, and calculus, including gradient, divergence, and curl. Learn about unit vectors and their properties.
