(Minerva Uni) (3Blue1Brown) An object that has both magnitude/length and direction. In space, it’s visualized as a directed line segment.
A vector
: tail, coordinate – head, coordinate - magnitude/length: can be calculated using Pythagorean theorem or dot product of itself
- direction:
, the angle between AB and the horizontal -axis
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- When we say algebraic vector, we care more about its representation as numbers.
Fundamental Operations
Given two vectors
Addition
It’s the same as adding two algebraic vectors. We say addition of vectors is commutative.
Scalar Multiplication
Multiplying by a scalar
If the scalar is negative, the vectors change to the opposite direction. dot product is a scalar, so
Angle between two vectors
See dot product
Dot product
dot product
dot product of two vectors is an operation that takes two equal-length sequences of numbers (vector
and ) and returns a single number
= ith component of vector = ith component of vector : dimension of the vector space or number of components of either vector Properties
Aside
A dot product
can be written as the product of the transpose vector with the other vector Distributivity & Associativity
dot product is distributive
and associative
Relationship with 2D vector
(3Blue1Brown) Intuitively, dot product linearly transform one of the vector
Find the magnitude of a vector
The magnitude of a vector is equal to the square root of the dot product of itself
Find the angle between vectors
The angle
between two vectors and is computed by their dot product over the product of vector lengths Find the angle for vector
and Prove orthogonal
Two vectors are orthogonal/perpendicular if and only if their dot product is 0.
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Example
(The Organic Chemistry Tutor)
Given two vectors