Vectors Flashcards
Is dot product distributive?
Yes
a(b+c) = ab + a*c
Is dot product associative?
No
a(bc) != (ab)c
What is a dot product?
Dot product is a way to multiply two vectors.
The result of the dot product is a scalar that represents relationship between two vectors
v*w = sum(vi + wi)
Give an algebraic definition of a vector
A vector is an ordered list of numbers (elements)
[1 2 3] != [3 2 1]
What is the term for the number of elements in a vector
Dimensionality
What is the dimensionality of a vector?
Number of elements in a vector
Column vectors vs row vectors
Column vectors are standing vectors:
| 1 |
| 2 |
| 3 |
Row vectors are “lazy” vectors or vectors that are lying down:
[1 2 3]
Geometric definition of a vector
A vector is a straight line with some length and some direction, specified by numbers in the vector.
What the starting point of the vector is called?
Tail
What the ending point of the vector is called?
Head
What’s the vector starting at the graph’s origin called?
A vector in standard position
Are two vectors with the same length and direction but with different starting points different or the same?
No, the starting point does not matter. As long as vectors have the same length and direction, they are considered identical.
What is the difference between a vector and a coordinate?
A coordinate is defined by numbers, and there can be only one point corresponding to it. A vector, on the other hand, is defined by its length and direction, meaning it can be positioned anywhere on the graph.
What a single number is called? And why?
Scalar, because vector-scalar multiplication (a vector multiplied by a scalar) can be think of as scaling vector up or down by given number (scalar)
In which special cases dot product is associative?
- When one of the vectors is zero vector
- When all vectors are equal
Is dot product commutative?
Yes
a * b = b * a
How to calculate the length of a vector?
The length of a vector (||a||) is equal to the square root of vector’s dot product with itself
||a|| = sqrt(a * a)
Two vectors have an angle <90 degrees. How would this affect the dot product?
The cosine will be >0 (positive)
The dot product will be >0 (positive)
Two vectors have an angle >90 degrees. How would this affect the dot product?
The cosine will be <0 (negative)
The dot product will be <0 (negative)
Two vectors have an angle =90 degrees. How would this affect the dot product?
The cosine will be =0
The dot product will be =0
Two vectors have an angle =0 degrees. How would this affect the dot product?
The cosine will be =1
The dot product will be >0 (positive)
The geometric formula for the dot product
The formula for the dot product α between vectors a and b with angle θ:
a = |a||b|cos(θ)
How vectors with a 90 angle between them are called?
Such vectors are called orthogonal
