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 direction 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