12-06: Geometric Vectors

Vectors

• Scalar: a quantity that @@describes magnitude or size only@@ (with or without units) @@but does not include direction@@
    * Magnitude, no direction
      * Magnitude: a number/numerical value is associated
    * e.g. mass (2 kg), speed (2 km/hr)
• Vector: a @@quantity that has both magnitude and direction@@
    * Magnitude and direction
      * Direction: where it is going
    * e.g. velocity (2 km/hr west)

Vectors can be represented as:

• The magnitude of a vector is designated using $absolute value brackets$

A vector’s direction can be expressed using many different methods

Parallel Vectors

• Parallel vectors: have the same or opposite direction, but not necessarily the same magnitude

   Equivalent vectors: have the same magnitude and the same direction

   Opposite vectors: have the same magnitude but opposite direction

• Multiplying a vector by a negative means that the direction gets flipped (or you could also say that its origin point gets flipped)

Addition & Subtraction of Vectors

When you add 2 or more vectors, you are finding a single vector called the ^^resultant^^

• Think of adding vectors as finding a shortcut, it accomplishes the same thing that the original vectors did when applied one after the other

Adding Vectors

2 methods to adding vectors:

Subtracting Vectors

• To subtract c-d, add the opposite of d to c

2 methods to subtracting vectors:

Parallel vectors:

If ^^2 vectors are parallel and acting in the same direction^^, the ^^overall magnitude^^ is equal to: the ^^sum of the individual vectors^^ (use simple addition)

If %%2 vectors are parallel but acting in opposite directions,%% the %%overall magnitude%% is equal to: the %%difference of the two individual vectors%% (use simple subtraction)

Zero vector: when two opposite vectors are added together, the resultant has zero magnitude and no specific direction. Written as

Properties of Vector Addition

Multiplying a Vector by a Scalar

• We can multiply a vector by a number k to produce a new scalar multiple of the vector

• k is used as multiplication and is called a scalar, it can be any real number
• %%Multiplying a vector by a scalar k can impact the vector’s magnitude & direction%%

Rules:

If %%k>0,%% then (vector) kv has the %%same directio%%n as (vector) v

If ^^k<0^^, then (vector) kv has the ^^opposite direction^^ as (vector) v

If 0<|k|<1, then the vector is decreased in magnitude, shortened

If ==|k| > 1, then the vector is increased in magnitude==, lengthened

If @@k=0, then the result is a zero vector@@

 Vectors that are scalar multiples are parallel and are said to be colinear. They form a straight line when arranged tip to tip

Vector properties for scalar multiples

• Distributive property

• Associate property

• Identity property

Applications of Vector Addition

Resolution

• @@Resolution: taking a single force and decomposing it into 2 components@@
• Most useful and important way to resolve a force vector occurs when this vector is resolved into 2 components that are at right angles to each other
• A vector can be resolved into 2 perpendicular vectors whose sum is the given vector

Rectangular components of vectors: 2 perpendicular vectors that are added to give a resultant

 Equilibrant vector: one that balances another vector/combination of vectors. Equal to the magnitude but opposite in direction to the resultant vector

• If the equilibrate is added to a given system of vectors, the sum of all vectors including the equilibrant is a zero vector