Adding Vectors at an Angle
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
|
|
|
|
 |
 |
 (link disabled) |
|
||
 |
 |
|
|||
 |
 |
|
|||
 |
 |
|
|||
 |
 |
 |
|
||
Webmaster (link disabled)
Almost
everything in motion will change its direction or movement pattern.
Cars can swerve, surfers twist and turn; these are such examples of
movement. The resultant displacement on a angle can be different
form the actual distance travelled. Even the most complex systems
of movement can be depicted on a vector diagram, which is why
Oliver Heaviside invented the vector diagram. The resultant
displacement and the distance travelled can be totally
different.
If you look at the above the picture you can see that the resultant displacement will be different from the total distance the yellow raft will travel. To find the resultant displacment you must have the direction, and distance travelled.
Direction
Direction is a very important aspect of investigating vectors. We use a common convention of direction, the compass. If a direction does not exactly match a direction on the compass then we write it as a angle from the closest compass point. To measure the angle exactly you must use a protractor. The direction can be written as [30o E of N]. The diagram for this description can look like the following:
Adding Two-Dimensional Vectors using Scale Diagrams
If you know the size and direction of each displacement then we can draw a scale diagram in order to obtain the resultant displacement.
|
Sample Cindy decided she wanted to visit her friend Kelvin. She decides to take a bus 30 km south to a near by street. She then walked 10 km west to his house. What is the resultant displacement? 1 cm = 10
km
This drawing was not created accurately, but it provides a example for this question. (you might try drawing this out on a piece of graph paper) If you measure this out, you will get a displacement of 32 m [70o west of south]
|
1 = 30 km [S]
Adding Two-Dimensional Vectors Algebraically
Using algebra to add two-dimensional vectors together is more complicated that using it for one-dimensional vectors; in fact you have to use the method for adding one-dimensional vectors within this method. Essentially, what you have to do is differentiate the vertical vectors from the horizonal vectors, add them together (respectively), and come up with two resultant vectors (one vertical and one horizonal). Apply the Pythagorean theorum to these to find the distance compontent of the final vector, and divide them to find the slope of the final vector (which can be converted to degrees as the direction of the final vector). If the vector has both a vertical and horizontal component, you have to use its slope to isolate these components.
However, adding two-dimensional vectors algebraically is not a grade 10 topic and will therefore not be covered in detail here; this summary is just meant to prepare you for the rest of high school.
Continue to the next lesson: Velocity