Head to Tail
It is important to draw vectors head to tail just like when following directions. To go to the mall you would not go back to the origin (beginning) before the next direction. The end of the last direction has an arrow tip head. The head of the last direction is where the tail of the new instruction begins every time.
Minimum and Maximum
When vectors are in the same plane, i.e. only X axis, the minimum displacement will be when they are in opposite directions. When vectors are facing the same direction the displacements add up. Pythagorean Theorem and Trigonometry functions SOH CAH TOA are used when vectors are not on the same plane. The rest of this page will show you how to do this.
Either Vector First
When adding two vectors, 20 meters north and 40 meters east, you would start with an origin dot then draw them head to tail. It does no matter which one you draw first as long as they are head to tail. The magnitude will be the same either direction you follow first but you'll have complimentary angles and descriptions. One may be 27° N of E and the other would be 63° E of N. Complimentary means the two add up to 90°.
Describing The Direction
Directions have a baseline, starting direction from which the angle is directed, described many possible ways.
- From the +X axis
- And Many Others
Look at the context of the problem to determine which to use.
The angle will be directed from that baseline and needs a direction as well. Look at the difference between 20° N of E and 20° N of W. Both are 20° north but without the baseline the direction is uncertain. Of east or of west is a baseline that completes the direction.
Determining Components of a Vector
What is the east component of 85 m 20° North of East?
When you have a question like this you are trying to find a component of a vector. Because of being at an angle north of east, this vector will have a north component and east component that make it up. Follow the directions to solve for the east component of this vector.
- Draw an origin dot
- From the origin draw the last direction you see, east in this description (85m 20° North of East). Be sure to include the arrow tip. This represents your east component.
- Continue head to tail, draw the second to last direction, north in this description (85m 20° North of East). Also include an arrow tip. This represents your north component.
- Next draw the resultant, our hypotenuse. Go back to the origin and draw the arrow from start to finish. This represents the original vector described.
- Now place the 20° angle closest to the origin dot (85m 20° North of East)
- Lastly place the magnitude, 85m, on the hypotenuse.
Now that you drew the vector, you have three arrows, one on the hypotenuse, one east, and one north. The north and east arrow represent components. We are asked to solve for the east, so we will use our trig functions to figure out that side. The arrow in red is facing east and what we are solving for. We have the 85 m hypotenuse, a 20° angle, and the adjacent side as our unknown. This leads us to the cosine function which rearranges to adj = (cos Ө)(hyp).
adj = (cos Ө)(hyp)
adj = (cos 20)(85) = 80 m
The final answer to the question rounds to 35 meters east.
Adding Two Vectors on Different Axis
What is the resultant of a horse that travels 45 meters north followed by 75 meters west?
When you see a question that has two separate vectors, 45 meters north, and 75 meters west, you are adding vectors. When adding two vectors on different axis, you can't simply add or subtract. Follow the following step as we go through an example to get the three part resultant answer. The answer will have a magnitude, an angle, and a description of that angle.
- Draw an origin dot
- Draw the first component (45 meters north) and label it
- Draw the next component (75 meters west) and label it starting at the previous arrow tip
- Go back to the origin and draw the resultant from beginning to where the components ended (See the blue line above).
- Use the Pythagorean Theorem to determine the hypotenuse.
- Use inverse tangent to figure out the angle.
- Describe the angle ____ of ____. The last direction is the baseline (line coming out of the origin dot) so W of N.
How do you describe this vector?
2. What is the west component of 67.2 m/s 22° north of west?
3. What is the resultant of 40 meters east and 22 meters south?
- To determine the magnitude of the hypotenuse you use the Pythagorean Theorem as seen above.
- To determine the angle with an opposite and adjacent we use the inverse tangent function as seen above.
- When describing the angle it will be from the baseline direction "of North"
The final answer here is 87 m 59° W of N. The magnitude is 87 m, the angle is 59°, and the description is to the west of north.
2D Non Projectile Motion Quiz