Kinematics in Two or More Dimensions (INCOMPLETE)

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Independence of Motion in 2+ Dimensions

Vectors can move in any direction, these directions are independent of each other!

Vectors in 2+ Dimensions

A vector has magnitude AND direction

A vector can be decomposed into its components based on the coordinate system (we use standard X-Y)

They MUST be orthogonal (perpendicular to each other)

All the rules of physics still apply, but now we do it per component

Tip to Tail Concept

Vector A and Vector B may exist anywhere on a graph

If you move the STARTING POINT (tip) of Vector B to Vector A's tail, you may create a third Vector C to connect them to find the length of A + B

In other words, A + B = C

NOTE: THE NEGATIVE OF A VECTOR POINTS IN THE OPPOSITE DIRECTION!

EX: A = ---------->, -A = <------------

Analytical Understanding of the Tip to Tail Concept

Subtracting a vector is the same as adding its negative

When we move Vector B to Vector A, if Vector B is meant to be subtracted, it will move in its opposite direction, still at the starting point of the tail of Vector A

Mathematical Understanding of the Tip to Tail Concept

Now we can understand, we can say that any vector can be considered a sum of two vectors

AsubX + AsubY = A

Remember, AsubX and AsubY are independent of each other!

To know more, we can look at the angle that the vector creates to the X axis, we will be using sin, cos, and tangent!

SOH CAH TOA

We also need to know how long A itself is, for this we may use the pythagreon theorum

a^2 + b^2 = c^2

theta = angle C

a = b cos theta

c = b sin theta

c / a = tan theta

Summary

Magnitude of Vector A = √(AsubX ^2 + AsubY^2)

Magnitude of Vector Components, use sin and cos that involve Vector A!

Interior Angles: theta = inverse tan (AsubY/AsubX)

Adding or Subtracting Vectors in XY