Chapter 6 - Scalars and Vectors

6.1 Scalar and Vectors

Scalars

  • Magnitude

  • Only magnitude and units are needed to understand the journey


Examples;

  • time

  • distance

  • volume

  • speed

  • temperature


Vectors

  • Magnitude + direction

  • Magnitude, direction and units to make sense


Examples;

  • position

  • displacement

  • velocity

  • acceleration

  • force

  • momentum


Vectors as Arrows

In a vector diagram

  • Length indicates magnitude

  • Arrowhead shows the direction (tail to head)

  • Force - push or pull and is measured in the unit Newtons (N)

  • Exact scale is not always accurate, but are drawn relative to another.


Point of Application of Arrows

If vector represents a force, the tail end is draw at the point where force is applied to object

If it is a displacement vector, the tail is attached to the position where the object starts.

Friction vectors are drawn at the point where they act between an object and a surface.


Direction Conventions

  • Vectors need direction


In One Dimension (1D)

  • forwards/backwards

  • Up or down

  • left or right

OR

  • north or south

  • east or west


Sign Convention

  • Used to covert physical directions to positive and negative

  • + ; forwards and right

  • - ; backwards and left


Horizontal Plane

Full Circle (True) Bearing

  • North is 0˚ T

  • Moves clockwise from north

  • Eg, 95° T is 95° clockwise from north


Quadrant Bearing

  • All angles are referenced from north and south and between 90˚ east or west

  • 30° T becomes N 30° E, which can be read as ‘from north 30° towards the east’.


Vertical Plane

  • 2D plane

  • Direction is references as upwards and downwards (vertical) or left and right (horizontal)

6.2 Adding Vectors in One and Two Dimensions

Adding vectors that are 1D means finding the resultant vector for a number of vectors that are in the same line

Adding Vectors in One Dimension

  • Adding two or more vectors in the same dimension = vectors are pointing in same direction OR opposite direction (collinear)


Graphical Method of Adding Vectors

  • Sum of vectors (resultant vector) is drawn from the tail of the first vector to the heat of the last vector.


Algebraic Method of Adding Vectors

  • A sign convention is used to represent direction

  • Allows you to enter the signs and magnitudes into a calc


Adding Vectors in Two Dimensions

  • Vectors must be in in the same plane (coplanar)

  • Go in any direction within the plane and can be separated by an angle (right angles, acute or obtuse)

  • Direction Conventions; north, south, east, west OR forwards, backwards, left, right


Graphical Method of Adding Vectors

Magnitude and direction of a resultant vector can be found by drawing a scaled vector diagram using

  • Head to tail method

  • Parallelogram method


Head to Tail Method

  • Use appropriate scale

  • Add vectors at right angles head to tail

  • Measure length of resultant vector and compare it to scale

  • Describe and measure the direction apporpriately

  • If vectors are at 90˚ to each other, the graphical method is better


Parallelogram Method

  • The two vectors to be added are drawn tail to tail

  • A parallel lone is drawn for each vector

  • The resultant vector is drawn from the tails of both to where the head meet


Geometric Method of Adding Vectors

  • Only give approximate results

  • More accurate using pythag and trig

6.3 Subtracting Vectors in One and Two Dimensions

Subtracting Vectors in One Dimension

Need → Final vector - Initial Vector

1) Work out which is initial vector

2) Reverse its direction

3) Add these two vectors together (final vector and opposite of initial vector)


Graphical Method of Subtracting Vectors

  • Velocity gives indication of how fast an object is moving in a certain direction

  • It is vector because the direction is important when saying the velocity


1) Determine which vector is initial and which is final

2) Final velocity is drawn first

3) Initial is then drawn but in opposite direction from its original form

4) The resultant vector is drawn from the tail of the final to the head of reversed initial


  • Magnitude from resultant vector can be calculated from the magnitudes of the two vectors


Algebraic Method of Subtracting Vectors

  • Sign convention used

Equation = Δv = v2 + (−v1)

change in velocity = final velocity - initial velocity


Subtracting Vectors in Two Dimension

  • Initial velocity must always be reversed before it is added to the final velocity

Graphical Method of Subtracting Vectors

1) Draw the final velocity first

2) Draw the opposite of the initial velocity head to tail with the final velocity vector

3) Draw the resultant change in velocity vector, starting at the tail of the final velocity

vector and ending at the head of the opposite of the initial velocity vector.

4) Measure the length of the resultant vector and compare it to the scale to

determine the magnitude of the change in velocity.

5) Measure an appropriate angle to determine the direction of the resultant vector.

Geometric Method of Subtracting Vectors

  • Only gives approximate results as it relies on comparing magnitude of the change in velocity to a scale and measuring with a protractor

  • Use Pythagoras and trigonometry for accuracy

6.4 Vector Components

  • Trigonometry


Finding Perpendicular Components of a Vector

  • two components that are at right angles to each other, when added together get the original vector.

  • To find the components of a vector, a right-angled triangle is constructed with the original vector as the hypotenuse.


Geometric Method of Finding Vector Components

  • Construct a right angled triangle using OG Vector as hypotenuse

  • Magnitude and direction are determined using trig