We are already familiar with some scalar quantities such as mass (6 kg), time (3 s) and distance (2 km). After examining the similarities in these examples you should notice that scalar quantities consist of a magnitude ( a number and a unit). In order to add scalar quantities your units must be the same.
i.e. if you were to add 12 min to 2 hr you would have to convert hours to minutes or vise versa.
On the other hand, vector quantities consist of a magnitude and a direction (32 m [N]). Because of the need for a direction, vector quantities are able to describe various aspects of kinematics and dynamics. Position, displacement, and velocity are all examples of vector quantities.
Vectors are used to represent vector quantities on a diagram. A vector is composed of a line segment drawn to scale with an arrowhead at one end. The tail of the vector is at its origin and the tip is at the terminal point (the arrowhead). The length of the vector represents its magnitude and the arrowhead indicates its direction. When drawing vectors you must also include reference coordinates.
Collinear vectors are vectors that exist in the same dimension. In other words, they exist either in the same direction or in the opposite direction.
Special types of collinear vectors are equivalent vectors. These vectors are equal in magnitude and direction.
The direction of a vector is always expressed in square brackets in one of two forms.
One of the four basic directions: north, south, east, or west.
Relative to a north or south position such that the degrees is less than 90.
Note: The tail is always located at the origin of an imaginary Cartesian plane
Vectors can be added algebraically if the directions are the same or if you can make them the same with the addition of a minus sign.
6 cm [W] = -6 cm [E]
The negative vector quantity indicates that its direction is actually opposite to that stated.
In this example, the object is actually moving 6 cm west.
The vector sum of two or more vectors is called the resultant vector. Now, determine the resultant vector for the following.
7 km [E] + 10 km [E] + 8 km [W]
=7 km [E] + 10 km [E] - 8 km [E]
= 9 km [E]
The resultant vector is 9 km [E]
Vectors can also be added graphically on a neat, accurate and properly scaled diagram using the tail to tip method of addition.
Step1: Decide on a scale and draw your reference coordinates to the right of your page.
Step 2: Indicate the starting point of your first vector with an x.
Step 3: Draw one of the vectors placing its tail at the x. Remember to label all vectors with a magnitude and direction.
Step 4: Draw your next vector starting at the tip of the last vector. Continue in this manner until all vectors have been drawn.
Step 5: Draw a dotted line form the x to the terminal point of your final vector. This new vector represents the vector sum and is called the resultant vector.
Step 6: Measure the resultant vector and determine its direction from the starting point (x) using a protractor.
It is important to remember that the vectors can be added in any order and that this method can be used to add any type and number of vectors.
Using the steps outlined above, graphically determine the sum of the following vectors.
20 km [N], 12 km [S], and 14 km [N] *Click Start to watch the vectors add.
Scale: 1cm = 2 km
: 20 km [N]
: 12 km [S]
: 14 km [N]
: which is 22km [N].
Discuss with a classmate whether or not temperature is a scalar or vector quantity.