Vectors
What is an arrow?
Connects two points in R^n in some order
Eg. P -> Q moves from point P to point Q
Let u = [u1, u2, ..., un] and v = [v1, v2, ..., vn] be vectors in R^n. What is u + v?
u + v = [u1 + v1, u2 + v2, u3 + v3]
What is a norm?
The magnitude/length of a vector
What are the two things needed to form a vector equation of a line?
A point and a direction vector
What is a linear combination?
A series of vectors all of which are scaled by constants to retrieve a resulting vector
Given two arbitrary points, ie. P = (p1, p2, p3) and Q = (q1, q2, q3), how is P -> Q represented?
P -> Q = (q1 - p1, q2 - p2, q3 - p3)
How do you add two vectors geometrically?
Net journey from the tail of one vector to the tip of the other
How do you normalize a vector?
Multiplying a vector by one over its magnitude
What does the point in a linear motion equation represent?
The position of motion when t = 0
What are the standard unit vectors in R^3?
e1 = [1, 0, 0]
e2 = [0, 1, 0]
e3 = [0, 0, 1]
What is a vector?
A position-less arrow
Arrows in different positions can correspond to the same vector if their lengths and directions are the same
What happens when you multiply a vector by some scalar?
Every component in that vector is scaled by that number
When is the norm of a vector equal to zero?
If and only if the vector in question is the zero vector
What is are parametric equations?
Equations in a variable that govern the coordinates of a point in the space.
How do you prove or disprove a "If..., then..." statement?
To prove: Assume the "If" part is true and then work to derive the "then" part
To disprove: find a counterexample for which the "If" part is true and the "then" part is false
When is a vector in standard position?
When its tail is located at the origin
T/F: You can divide vectors.
F
What is another way to write the distance between two vectors u and v for it can be denoted as d(u, v)?
||u - v||
Find the vector function that describes the linear motion of a particle that passes through P(1, 2, -3) and Q(7, -2, 1) at t = 5.
x = [1, 2, -3] + t[6/5, -4/5, 4/5]
What is the span?
The set of all possible linear combinations for a given set of vectors
T/F: Every point in R^n corresponds to a vector in standard position and vice versa.
T
What makes two vectors parallel to each other?
One vector is the scalar multiple of the other
State the triangle inequality.
||u + v|| ≤ ||u|| + ||v||
Find the vector function that describes the linear motion of a particle that passes through P(1, 2, -3) at t = -2 and Q(7, -2, 1) at t = 5.
x = [19/7, 6/7, -13/7] + t[6/7, -4/7, 4/7]
What is the span of one vector?
A line