Vectors in 2D

Vectors in 2D

Vectors have a direction and magnitude. They can be added geometrically or algebraically. Geometrically, they are plotted on a graph, drawn as an arrow, and added using the "tip-to-tail" method. Adding and subtracting vectors requires simple operations on the coordinates.

Vector Addition Add vectors as you would coordinates. Example: What is <5,2> + <-3,1>? Add the components: <5+(-3), 2+1> = <2,3>

Vector Subtraction Same as addition, but with - . Example: What is <4,2> - <1,4>? Subtract the components: <4-1,2-4> = <3,-2>   To subtract geometrically, add the negative of the vector, which goes in the opposite direction. "Tip-to-tip".

Scalar Multiplication We can multiply a vector by any real number. Example: What is 3<1,5>? Multiply the components inside the vector by the real number 3: <3*1, 3*5> = <3,15>   Geometrically, we get <1,5>, but now it is 3 times as long as it originally was.   In general, when multiplying by a scalar, we get cv, where v is a vector, but c times as long. If c is negative, v points in the opposite direction, c times long. If c is zero, we get a zero vector: 0=<0,0>

Magnitude The length is the hypotenuse of a right triangle if we break the vector down into its components. So the magnitude of the vector becomes: a^2+b^2=c^2. Example: Find the length of <3,4>. Use the formula: sqrt(3^2+4^2) = sqrt(25) = 5.   In general, the length of a vector <a1, a2> is: |<a1,a2>| = sqrt((a1)^2+(a2)^2) In more dimensions, add another squared component: |<a1,a2,a3>| = sqrt((a1)^2+(a2)^2+(a3)^2)

Unit Vector A unit vector is a vector with a magnitude (length) of 1 unit. A unit vector can point in the same direction as another vector, but is scaled down to be size 1. Example: Find a unit vector in the same direction as <7,-24>. -First we need to find out how long the vector is. Use the magnitude formula: sqrt(7^2(-24)^2)= 25 units. -To shrink, we scale down by 1/length, or 1/25. -The unit vector becomes <7/25,-24/25>. 

Standard Unit Vectors <1,0> <0,1> <1,0,0> <0,1,0> <0,0,1>  

Properties of Vectors

1. Commutativity: a + b = b + a 2. Associativity: a + (b + c) = (a + b) + c 3. Identity: 0 + a = a 4. Negation: a + (-a) = 0 5. Distributivity: c(a + b) = ca + cb                               (c+d)a = ca + cb                               (cd)a = c(da) 6. Unity: 1a=a

Relevant Equations

Magnitude sqrt(a^2+b^2) = length, 2D sqrt(a^2+b^2+c^2) = length, 3D

Unit Vectors To scale down to 1: 1/(sqrt(a^2+b^2)), 2D 1/(sqrt(a^2+b^2+c^2)), 3D