Keo's Notes

Vectors

16 min read

Geometry, Linear-Algebra

Vector Spaces

A vector space is a set of objects with two operations and where is a field (generally or ). It must also follow 8 axioms For

  1. Additive Closure
  2. Additive Commutativity
  3. Additive Associativity
  4. Additive Identity
  5. Additive Inverse
  6. Multiplicative Closure
  7. Multiplicative Associativity
  8. Multiplicative Identity
  9. Scalar Distributivity
  10. Vector Distributivity

Common examples of vector spaces are the set of all matrices over .

Basis Vectors

Given a vector space defined over , a basis (also called the basis vectors) is a set of vectors that are linearly independent over with the property that every vector in can be written as a linear combination of the basis vectors over . If there are infinite elements in then there are infinitely many basis vectors.

Linear Independence

Given vectors the are linearly independent if no vector can be written as a linear combination of the others. The more rigorous definition is that a set of vectors is linearly dependant if there exists solution to the equation such that are not all zero.

If no non-trivial solution exists then the vectors are said to be linearly independent. This equation relates back to the idea of a vector being able to written as a linear combination of the others as if a solution exists then so a linear combination. You can also work backwards from the idea of a linear combination to get the equation. Also note that if then the vectors are linearly Dependant.

Span

Given a subset of vector space , is defined as the set of all finite linear combinations of the elements in . For two 2D non-zero vectors is the whole plane it the are linearly independent and if then it is a line containing . In general the span of , -dimensional linearly independent vectors is the full D space.

Norm

The norm of a vector denoted (or sometimes just ) is an idea of length and it must follow the following three axioms

  1. Non-Negativity
  2. Zero Vector
  3. Homogeneity
  4. Triangle Inequality

These all stem from Euclidean geometry and are properties that we would want to meaningfully talk about length.

Multiplying Vectors

Inner Product

There is no single canonical way to multiply vectors in fact there are multiple ways. When multiplying vectors we want to take into account length and angle. When multiplying vectors we want the result to be large and positive if they are close, 0 if they are perpendicular and negative if they are somewhat opposite so we can think of multiplication as measuring how “similar” two vectors are. We also want to have similar properties as regular multiplication so we have the inner product that follows the axioms

  1. Symmetry with conjugation (Note for we get pure symmetry )
  2. Linearity in the first slot
  3. Positive-definiteness and Notes: if we have pure symmetry then we also get linearity in the second slot and the reason for the conjugation for symmetry is to keep positive definiteness which is important as should be thought of as length squared which should have positive-definiteness.

For equal vectors it makes sense to just choose their length squared so we get the property that and there is only one such inner product that satisfies this. For a vector space over it is

Proof Using this we get our inner product as This product does satisfy all 3 inner product axioms but I won’t show all the details

Note that is not required and you can define the norm and inner product independently but this is a natural axiom to state as it follows from basic Euclidean geometry and gives rise to many useful properties. It also makes the inner product unique to the norm.

Dot Product

In Euclidean Geometry of we choose the norm which is the notion of length to be which is simply the Pythagorean theorem. Using the inner product in terms of the norm we get that Giving us the familiar dot product denoted as . We can also derive it geometrically. The dot product of is equal to projecting onto and multiplying the lengths and the new projected vector. It is signed so it the projection is pointing the opposite way of than the dot product is negative. Using some simple Trig we find that

The dot product is commutative. This arises from the fact that projections scale linearly. If had the same length then you can use symmetry to get the it is commutive in this scenario. Say you scale to get , then as the project of remains the same but then length of has scaled by . But if you consider then the length of the projection of scales by yielding . With any pair of vectors you can always take out some scaling factor to get this form of with the same length meaning for all vectors the dot product is commutative.

The dot product is also distributive over vector addition meaning for any length vectors Proof Hence .

Dual Vector

We also find that the dot product is equivalent to a linear transformation by the first vector as a matrix e.g.

And this generalises to any -length vector. 2D Example Proof To see why we can imagine placing a copy of the real number line on the 2D plane (rotated at any angle) with the condition that 0 lies on the origin. Then define to be the vector from the origin to where 1 lies on this number line. Then define a function that takes any 2D vector and projects it onto this number line with the number it lands on as output . Since this is a linear function we can find a matrix to describe this transform. By considering and using a line of symmetry we can find that this is . Applying this transform is computationally equivalent to taking the dot product. If we have a non-unit vector we can simply write it as and the dot product scales linearly so it still works.

This gives rise to the idea of a Dual Vector, that is given any linear transform which output is the number line you can find a vector such that taking the dot product with that vector is equivalent to applying the transform.

Vector Cross Product

The cross product of two vectors is defined as vector that has magnitude equal to that the signed area of the parallelogram define and direction perpendicular to both following the right hand rule mentioned later. It is only defined in 3 (and 0,1,7) dimensions.

We can compute the magnitude using the Determinant taking to be the column vectors of a matrix and computing . This is why the cross product is signed. This is because the determinant measures the factor that area scales. Since as they bound a unit square. are simply after a transformation under so the determinant is the area we are looking for. It is positive if is on the right of and negative if it’s on the left.

To compute the cross product fully we can take the determinant of a special Matrix

Note that this is equivalent to as determinant doesn’t change under a transpose. Also note that putting unit vectors as entries in a matrix is a notational shorthand.

Explanation First we define a function

This function geometrically is computing the signed volume of the parallelepiped bounded by This function is and it is linear since the Determinant is multi-linear and we are changing a single slot. Since it is a linear transform we can define the function as matrix multiplication and more specifically since it is a transformation to there must exist a dual vector

Meaning we can compute by taking this dot product so we get

By comparing coefficients of we get the coordinates of and writing it as a sum of a linear combination of unit vectors gets the exact same thing as the determinant as the matrix with unit vectors as entries.

We can think of this as asking for any two given vectors , what vector has the property that for any other vector , (projecting onto and multiplying the length of by the length of this projected vector) is equivalent to the volume of the parallelepiped bound by . As we have just shown is given exactly by . We can also think about this by finding the volume of the parallelepiped geometrically. First find the area of the base which is the area of the parallelogram bounded by then multiply by the component of which is perpendicular to both . But this is equivalent to taking the dot product of a vector perpendicular to as the dot product is itself a projection. There are two such vectors that satisfy this property (going in opposite directions) which is the cross product we are looking for is the one such that when the dot product is negative so is the volume of the parallelepiped by the right hand rule is negative.

Using this idea of the vector that is the vector with length equal to the area of the parallelogram we can find using some basic Trig that the are is equal to where is the angle between such that . Then we just want a vector perpendicular to both that follows the right hand rule. We call this normal vector which has magnitude 1 to preserve the length of .

It is Distributive

is the angle between and where . And is the unit vector perpendicular to both and

Right Hand Rule

The direction of depends on the right-hand rule, swapping and means that therefore , this means it is anti-commutative

Finding Volumes - Scalar Triple Product

This is simply the definition/what a Determinant represents and using the geometrical definition of the cross product.

Straight Lines

Colinear: Points are said to be collinear if they all lie on the same straight line

The equation of a straight line vector passing through point and is parallel to vector is where is the position vector of a general point on the line. So by taking varying values of you can find the position vectors of varying points that lie on the straight line. The line has cartesian equation

Suppose is the position vector of a point on a line and the line is parallel to . Let be the position vector of a general point on the line. so . Since is parallel to , so

Planes

Coplanar: Points are said to be coplanar if they all lie on the same plane

The vector of a plane that through position vector is: where r is the position vector of a general point on the plane with being the position vector of a fixed arbitrary point on the plane and are non-parallel, non-zero vectors on the plane The intuition of this is that gets you from the origin onto the plane and allow you to move across the plane

The cartesian equation of a plane is where the normal vector to the plane is . You can find the cartesian equation given the vector equation using the cross product

Suppose a plane passes through and has normal vector and let be an arbitrary point on . Then . Since lies on , so where is a constant so . This can also be written as . Where represents the Perpendicular Distance from the origin to the place if

Angles Between Vectors and Planes

The Acute angle between 2 intersecting Straight Lines (with direction vectors ) is given with the formula

The Acute angle between Straight Line (with direction vectors ) and Plane (with equation ) is given with the formula

The Acute angle between 2 intersecting Planes (with equations vectors and ) is given with the formula

Points of Intersection

To find if 2 lines and intersect set them equal and solve the system of equations, if solutions for and exist then they intersect, if not then there is no intersection. Two lines are skew if they do not intersect but are parallel meaning solutions for and don’t exist but their direction vectors are scalar multiples of each other.

To find the intersection of a plane and a line have the plane in the form then sub in the vector equation into the plane equation.

Perpendicular Distances

To find the shortest/perpendicular distance between 2 lines and , Let be a point on and be a point on then shortest/perpendicular distance between and will be the straight line segment that is perpendicular to both lines. So you can then solve for and To find shortest distance between a line and a point , Let be a point on then . You can then find the coordinates of so you can find the distance The perpendicular distance from the origin to the plane is simply k The perpendicular distance from a point to the plane is

The distance between two parallel planes and is

Direction of Cosines

If a line is parallel to the vector the direction ratios are and the direction of Cosines of the line are

The sum of the squares of the direction cosines is always one

Graph View

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