12-06: Geometric Vectors

# Vectors

**Scalar**: a quantity that describes magnitude or size only (with or without units) but does not include directionMagnitude, 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 directionMagnitude 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 directionMultiplying 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

RULE | FORMULA | WHEN TO USE |
---|---|---|

Pythagorean | a² + b² = c² | Right triangles, when given 2 sides and looking for a 3rd side |

SOH CAH TOA | sinϴ = opp/hyp, cosϴ = adj/hyp, tanϴ=opp/adj | Right triangles, given an angle and a side & finding a side |

Sine law | a/sinA = b/sinB = c/sinC | Non right triangles, 2 sides and opp angle finding angle |

Cosine law | a² = b² + c² - 2bccosA | Non right triangles, given 2 sides and enclosed angle finding 3rd side |

### 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

# 12-06: Geometric Vectors

# Vectors

**Scalar**: a quantity that describes magnitude or size only (with or without units) but does not include directionMagnitude, 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 directionMagnitude 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 directionMultiplying 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

RULE | FORMULA | WHEN TO USE |
---|---|---|

Pythagorean | a² + b² = c² | Right triangles, when given 2 sides and looking for a 3rd side |

SOH CAH TOA | sinϴ = opp/hyp, cosϴ = adj/hyp, tanϴ=opp/adj | Right triangles, given an angle and a side & finding a side |

Sine law | a/sinA = b/sinB = c/sinC | Non right triangles, 2 sides and opp angle finding angle |

Cosine law | a² = b² + c² - 2bccosA | Non right triangles, given 2 sides and enclosed angle finding 3rd side |

### 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