Vectors in 2D

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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>

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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

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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