**Where are we going with this?**The information on this page relates to the skills needed to investigate and evaluate the graphical and mathematical relationship (using either manual graphing or computers) of one-dimensional kinematic parameters (distance, displacement, speed, velocity, acceleration) with respect to an object's position, direction of motion, and time.

**Understanding Vectors and Vector Math**

*(Oh brother! What now?)*

*Yeah…*

The study of objects in motion and forces acting upon them relies heavily on using vectors. Thus, having an understanding of vectors is very important. So… we should talk about this…Avectoris a concept connected to quantities that specifies aAnswering many physics questions about the world around us best done by using vectors.magnitudeand adirection.

**Representing Vectors**

It is very, very common to

**draw diagrams showing how vector quantities are acting on an object.**Usually,

**there will be some sort of reference**

*thing*… the horizon… the plane of a ramp…

*often, the x-y axis*thingy.

*(Thingy? Really? That's the best you can do?)*

**Vectors, in the diagrams, are represented by arrows.**

**The direction that the arrow points indicates the vector's… direction.****The length of the arrow represents its magnitude.**

Vector arrows have a tail and a head. The pointy end is the head.

**the length of the arrows should be drawn exactly in proportion to their lengths**. So, a vector with a magnitude of 4 would be twice as long as one with a magnitude of 2 and half as long as one with a magnitude of 8.

*NOTE: any illustrations on this page are*

**NOT**strictly depicted!**angles should correctly represent the angle at which the vector operates**on the object being observed.

**Working With Vectors: The Big Picture**

**When more than one vectors of any give quantity (e.g. force, velocity, acceleration) operate on an object within some frame of reference, their effect combines**. The the vector that results in the combination of each of the component vectors.

Component vector:Any individual vector having magnitude and direction within a given frame of reference.Component vectors can be combined into aresulting vector(or theresultant vector).

A resultant vector can be broken down into its component vectors.

**Working With Vectors: Graphical Conceptualization**

**IF THEIR ANGLE WITHIN THE FRAME OF REFERENCE DOES NOT CHANGE**, be moved so that its tail connects to head of the second vector. All of the vectors in the system can be moved in this same manner. This only works if the moved vector remains parallel to its original direction.

*(That is the same as saying that the angle does not change.)***A line from the tail of the first vector to the head of the last vector represents the resultant vector.**

*Moving a vector so that its origin is in a different place, but without changing its direction or magnitude is called*

**vector translation**.

*Finding a resultant vector by translating component vectors is a process that is sometimes called*"

**The Triangle Method**."

The blue vector is translated to the end of the green vector, andthe red vector is the resultant vector. |

**The order in which the vectors are rearranged does not matter. (This will be true when we look at the math process, too.)**This is true combining vectors is essentially adding their effects, and since there's some fancy name

*(the commutative property)*for math that says it doesn't matter what order we do addition.

Source |

**if vectors point in opposite directions, their effects reduce each other**, even to the point of cancelling totally. This is easily observed when two vectors are operating on a single line.

**when operating in opposite directions, one vector can be thought of as positive and the other negative.**

**Working With Vectors: Math Processes**

**Vectors On One Line**

*A box moves from a point of origin 10 meters left, then 4 meters right, then 2 meters left, then 3 meters right. Where is the box in relationship to the point of origin.*

∆d = d1 + d2 + d3 + d4Where left is positive and right is negative,d1 = 10 md2 = -4 md3 = 2 md4 = -3 m∆d = (10 + (-4) + (2) + (-3)) m∆d = 5 m left

**Vectors At Right Angles To Each Other**

**The resultant vector will be the hypotenuse of the triangle formed.**

- Its magnitude can be found using the Pythagorean Theorem.
- Its angle can be found using arctan (inverse tangent).

**Once you have two perpendicular vectors, you can find the resultant**(as above).

MORE TO COME!