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?
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A mathematical object with both magnitude and direction, represented as an ordered list of numbers (components).- When are two vectors equal?
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When all corresponding components are equal.- Define the zero vector.
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The vector with all components equal to zero, denoted 0; it has no direction.- How do you add two vectors?
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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.
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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?
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Subtract corresponding components: u - v = (u1-v1, u2-v2, ..., un-vn).- Define the magnitude (or norm) of a vector.
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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?
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A vector with magnitude 1; obtained by dividing any vector by its magnitude: u = v / ||v||.- Define the dot product of two vectors.
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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?
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Use u·v = ||u|| * ||v|| * cos(θ), then solve: θ = arccos(u·v / (||u|| * ||v||)).- When are two vectors orthogonal?
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When their dot product equals zero: u·v = 0 (they are perpendicular).- What is the geometric meaning of the dot product?
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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.
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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?
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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?
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Cross product is anti-commutative: u x v = -(v x u); dot product is commutative: u·v = v·u.
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