Vectors

Question 1

Is dot product distributive?

Answer 1

Yes
a(b+c) = ab + a*c

Question 2

Is dot product associative?

Answer 2

No
a(bc) != (ab)c

Question 3

What is a dot product?

Answer 3

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)

Question 4

Give an algebraic definition of a vector

Answer 4

A vector is an ordered list of numbers (elements)
[1 2 3] != [3 2 1]

Question 5

What is the term for the number of elements in a vector

Answer 5

Dimensionality

Question 6

What is the dimensionality of a vector?

Answer 6

Number of elements in a vector

Question 7

Column vectors vs row vectors

Answer 7

Column vectors are standing vectors:
| 1 |
| 2 |
| 3 |
Row vectors are “lazy” vectors or vectors that are lying down:
[1 2 3]

Question 8

Geometric definition of a vector

Answer 8

A vector is a straight line with some length and some direction, specified by numbers in the vector.

Question 9

What the starting point of the vector is called?

Answer 9

Tail

Question 10

What the ending point of the vector is called?

Answer 10

Head

Question 11

What’s the vector starting at the graph’s origin called?

Answer 11

A vector in standard position

Question 12

Are two vectors with the same length and direction but with different starting points different or the same?

Answer 12

No, the starting point does not matter. As long as vectors have the same length and direction, they are considered identical.

Question 13

What is the difference between a vector and a coordinate?

Answer 13

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.

Question 14

What a single number is called? And why?

Answer 14

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)

Question 15

In which special cases dot product is associative?

Answer 15

1. When one of the vectors is zero vector
2. When all vectors are equal

Question 16

Is dot product commutative?

Answer 16

Yes
a * b = b * a

Question 17

How to calculate the length of a vector?

Answer 17

The length of a vector (||a||) is equal to the square root of vector’s dot product with itself
||a|| = sqrt(a * a)

Question 18

Two vectors have an angle <90 degrees. How would this affect the dot product?

Answer 18

The cosine will be >0 (positive)
The dot product will be >0 (positive)

Question 19

Two vectors have an angle >90 degrees. How would this affect the dot product?

Answer 19

The cosine will be <0 (negative)
The dot product will be <0 (negative)

Question 20

Two vectors have an angle =90 degrees. How would this affect the dot product?

Answer 20

The cosine will be =0
The dot product will be =0

Question 21

Two vectors have an angle =0 degrees. How would this affect the dot product?

Answer 21

The cosine will be =1
The dot product will be >0 (positive)

Question 22

The geometric formula for the dot product

Answer 22

The formula for the dot product α between vectors a and b with angle θ:
a = |a||b|cos(θ)

Question 23

How vectors with a 90 angle between them are called?

Answer 23

Such vectors are called orthogonal