Vectors Flashcards
(9 cards)
What is a vector?
A vector is an element of a vector space, represented as an ordered list of numbers indicating magnitude and direction.
What does it mean for two vectors to be linearly independent?
Two vectors are linearly independent if one cannot be expressed as a scalar multiple of the other.
How do you calculate the Euclidean (L2) norm of a vector v = [x, y]?
‖v‖₂ = √(x² + y²)
How do you calculate the L1 norm of a vector v = [x, y]?
‖v‖₁ = |x| + |y|
What does the dot product of two vectors represent?
The dot product measures how much one vector projects onto another. If the dot product is zero, the vectors are orthogonal.
What is the formula for the dot product of u = [x₁, y₁, z₁] and v = [x₂, y₂, z₂]?
u ⋅ v = x₁x₂ + y₁y₂ + z₁z₂
How do you calculate the angle θ between two vectors?
cos(θ) = (u ⋅ v) / (‖u‖ ‖v‖), and θ = cos⁻¹(result).
How do you perform the cross product of two 3D vectors u and v?
Use the determinant formula:
u × v =
|i j k |
|uₓ uᵧ u𝓏|
|vₓ vᵧ v𝓏 |