Vectors, vector operations, and an intro to vector spaces from a college linear algebra course.
35 cards · basic cards · AI-written, checked twice. Edit anything.
- What is a vector?
- A mathematical object with both magnitude and direction, represented as an ordered list of numbers (components).
- When are two vectors equal?
- When all corresponding components are equal.
- Define the zero vector.
- The vector with all components equal to zero, denoted 0; it has no direction.
- How do you add two vectors?
- Add corresponding components: if u = (u1, u2, ..., un) and v = (v1, v2, ..., vn), then u + v = (u1+v1, u2+v2, ..., un+vn).
- Define scalar multiplication of a vector.
- Multiply each component by the scalar: if c is a scalar and v = (v1, v2, ..., vn), then cv = (cv1, cv2, ..., cvn).
- What is vector subtraction?
- Subtract corresponding components: u - v = (u1-v1, u2-v2, ..., un-vn).
- Define the magnitude (or norm) of a vector.
- The length of the vector; for v = (v1, v2, ..., vn), the magnitude is ||v|| = sqrt(v1^2 + v2^2 + ... + vn^2).
- What is a unit vector?
- A vector with magnitude 1; obtained by dividing any vector by its magnitude: u = v / ||v||.
- Define the dot product of two vectors.
- For u = (u1, u2, ..., un) and v = (v1, v2, ..., vn), the dot product is u·v = u1*v1 + u2*v2 + ... + un*vn.
- How do you find the angle between two vectors?
- Use u·v = ||u|| * ||v|| * cos(θ), then solve: θ = arccos(u·v / (||u|| * ||v||)).
- When are two vectors orthogonal?
- When their dot product equals zero: u·v = 0 (they are perpendicular).
- What is the geometric meaning of the dot product?
- It measures how much two vectors point in the same direction; equals the magnitude of one times the component of the other in that direction.
- Define the cross product.
- For vectors u and v in R^3, u x v is a vector perpendicular to both with magnitude ||u|| * ||v|| * sin(θ), where θ is the angle between them.
- What is the component formula for the cross product?
- For u = (u1, u2, u3) and v = (v1, v2, v3), u x v = (u2*v3 - u3*v2, u3*v1 - u1*v3, u1*v2 - u2*v1).
- How does the cross product differ from the dot product in commutativity?
- Cross product is anti-commutative: u x v = -(v x u); dot product is commutative: u·v = v·u.