Vectors

Question 1

What is a vector?

Answer 1

A vector is an element of a vector space, represented as an ordered list of numbers indicating magnitude and direction.

Question 2

What does it mean for two vectors to be linearly independent?

Answer 2

Two vectors are linearly independent if one cannot be expressed as a scalar multiple of the other.

Question 3

How do you calculate the Euclidean (L2) norm of a vector v = [x, y]?

Answer 3

‖v‖₂ = √(x² + y²)

Question 4

How do you calculate the L1 norm of a vector v = [x, y]?

Answer 4

‖v‖₁ = |x| + |y|

Question 5

What does the dot product of two vectors represent?

Answer 5

The dot product measures how much one vector projects onto another. If the dot product is zero, the vectors are orthogonal.

Question 6

What is the formula for the dot product of u = [x₁, y₁, z₁] and v = [x₂, y₂, z₂]?

Answer 6

u ⋅ v = x₁x₂ + y₁y₂ + z₁z₂

Question 7

How do you calculate the angle θ between two vectors?

Answer 7

cos(θ) = (u ⋅ v) / (‖u‖ ‖v‖), and θ = cos⁻¹(result).

Question 8

How do you perform the cross product of two 3D vectors u and v?

Answer 8

Use the determinant formula:
u × v =
|i j k |
|uₓ uᵧ u𝓏|
|vₓ vᵧ v𝓏 |