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Vector Questions, a selection of answers from the Dr. Math archives.

What is a Vector?
I am having trouble understanding exactly what a vector is and cannot seem to find a simple, straightforward explanation.

Definition of a Vector
I would like a "proper" definition of a vector and how concepts of "magnitude" and "direction" arise in the discussion of vectors.

Sum of Two Vectors
Why is the magnitude of the sum of two vectors less than or equal to the sum of the magnitudes of each vector?

Using Vectors in Geometry and Physics
How do you use vectors in problems about medians, areas, and acceleration and velocity?

Vector Algebra: Finding the Intersection Point
If I have two lines in three dimensions that I know intersect at some point, how do I work out what that point is? Both lines are defined by two points on each line.

Unit Vectors
Find the two unit vectors that are collinear with each of the following vectors. (a) vector A = (3, -5). I don't understand--is a unit vector only ever equal to 1?

Equation for Angle Formed by Two Vectors
Do I need to find the vector equation of OP and OQ, or of PO and OQ, when finding the angle POQ using the formula cos(theta)=a.b/ab?

Angle Between Vectors
Given vectors A and B in a plane, and vector C normal to that plane, compute the clockwise angle from vector A to vector B when viewed in the direction of the normal vector C.

Find the Velocity - a Vector Problem
A river flows westward at 8 m/s. A person wants to go directly across the river so that the resultant velocity is 12 m/s northward. Find the velocity of the motorboat that would be required to achieve this resultant velocity.

Vectors of Parallelograms and Octagons
ABCDEFGH is a regular octagon and AB = p and BC = q. Express AH in terms of p and q...

Vectors and Orientation
I am trying to write a game involving spaceships, and I have been having trouble finding relative positions.

Three-Dimensional Vectors
I am finding it very hard to understand and visualise the notion of a vector in 3 dimensions.

The Stationary Vector
In Markov processes, what does a stationary vector tell you and how do you find it?

Rotación de Véctores (3-D Vector Rotation)
Hay alguna fórmula para rotar véctores en el espacio sobre los tres ejes de coordenadas?

Intersecting Vectors and the Dot Product
Each of the following geometrical theorems can be proved with vectors, using the dot product...

Distance Between a Line and a Point Using Vectors
How do you use scalar vector projections to prove that the distance between the line ax+by+c = 0 and point (x1,y1) is |ax1+by1+c|/sqrt(a^2+b^2)?

Distance Between 2 Lines: Vectors
What is the shortest distance between 2 lines?

Vector Proofs

Vector Proof
Prove that given P, Q, R, and S (any 4 non-collinear points), with A and B the midpoints of PR and QS respectively, then PQ + RS = 2 AB...

Vector Proof: Parallelogram Diagonals
Use vectors to prove that the diagonals of a parallelogram bisect each other and the line joining the midpoints of two sides of a triangle...

Vector Space

Understanding Vector Spaces
What does a vector space in R^n mean? How can I prove that a list of numbers is a vector space?

Vector Spaces
How can I tell if V = {(x,y) in R^2 | y = 3x+1} a vector space when addition and scalar multiplication are defined by (x,y) + (x',y') = (x+x',y+y'-1) and k(x,y) = (kx,k(y-1)+1)?

Vector Space Dimensions
Why does C^2 considered as a vector space over the complex numbers have dimension 2, but as a vector space over the real numbers have dimension 4?

Basis for a Vector Space in R^3
Are the following bases for R^3: {(2,-3,1), (4,1,1), (0,-7,1)} ... ?

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