Download Notes on Vector Addition

Notes on Vector Addition
Purpose:
Relevance:
Success:
Name:____________________
Today I will learn vector components and how to add vectors together.
Vectors are used in almost every chapter of physics. Learning to properly add vectors will make many
problems much easier for you to solve.
I will be successful today if I can add horizontal, perpendicular, and odd angle vectors.
Basic Vector vocabulary

Vectors have a magnitude and direction.
Example: A velocity vector might look like this:
It could be written like this:


25 m/s
25 m/s, right or 25 m/s @ 0o or (25 m/s, 0o)
The positive X-axis is 0o. Positive Y-axis is 90o. Negative X is 180o. Negative Y is 270o.
When two or more vectors are added together, the answer to the problem is called the Resultant. The
resultant is also a vector and would include a magnitude and angle.
Adding Parallel Vectors


All the vectors being added are parallel to each other.
In that case the magnitudes add like integers.
Example 1
2.0 m/s
+
4.0 m/s
+
3.0 m/s
=
3.0 m/s
(Note that in the example above, right was considered positive and left negative, so the third vector’s
magnitude was subtracted. 2.0 + 4.0 – 3.0 = 3.0)
Adding Vectors at Right Angles



The vectors being added are at right angles to each other.
In this case, the vectors are treated as sides of a right triangle. The resultant would be the hypotenuse
of the triangle. The angle of the resultant is referenced to the positive X-axis.
We use the Pythagorean Theorem to find the hypotenuse and the tangent function to find the angle.
X2 + Y2 = R2
Example 1:
tan =
𝒀
𝑿
A river is flowing due east (0o) at a speed of 5.00 m/s. A boat is travelling due north (90 o)
across the river at a speed of 8.00 m/s. Find the resultant velocity vector of the boat.
Solution:
A diagram of the problem would look like this:
R
8.00 m/s

5.00 m/s
Solution Continued:
To find R:
R2 = 8.002 + 5.002
R = √89.0 = 9.43 m/s
To find :
tan = (8.00/5.00)
 = tan-1(1.60) = 58.0o
Answer:
9.43 m/s, 58.0o or 9.43 m/s @ 58.0o
Note: If the resultant is supposed to be in the 2nd or 3rd quadrants, then using tangent to find the angle will
give you 180o off from the angle you want. In a case like that, you add 180.000 o to your angle to get the
correct one.)
Finding X and Y Components

Every vector has what we call an “X component” and a “Y component”. If you think of a vector as the
hypotenuse of a triangle, then the X and Y components would be the two sides of the triangle.
Examples:
10.0 m
20.0 N
Y component
Y comp.
X component
X component

To find the X and Y components we use
X = Rcos
and
Y = Rsin
Where  is always measured from the positive X axis.
Example:
Find the X and Y components of the vector 44.5 m/s, 138o
X = 44.5cos138o = -33.1 m/s
Y = 44.5sin138o = 29.8 m/s
Example:
Find the X and Y components of 22.7 N @ -65o
X = 22.7cos-65o = 9.6 N
Y = 22.7sin-65o = -21 N
Adding Vectors that are NOT at right angles.




When adding vectors that are not at right angles, you can’t just use the Pythagorean Theorem and the
tangent of the angle.
The strategy is to find the X and Y components of all of the vectors being added. Then, add up all of
the X components to find the X component of the resultant. Add up all of the Y components to find
the Y component of the resultant.
THEN you can use the Pythagorean Theorem and tangent of  to find the resultant.
I recommend using the “Vector Grid” method. It helps keep the X and Y components straight, and it
helps with sig figs. The following is a vector addition using “The Grid”.
Example: Add the three force vectors listed below to find the resultant force.
F1 = 30.0 N @ 45.0o
F2 = 60.0 N @ 143.0o
F3 = 87.4 N @ 245o
Solution:
Step 1: Draw a grid that has 3 columns and enough rows for all three vectors plus the resultant.
Vector
X component
Y component
1
2
3
R
Step 2:
Fill in the X and y components of the three vectors using
X = Rcos
Vector
X component
Y component
1
30.0cos45.0o
30.0sin45.0o
2
60.0cos143.0o
60.0sin143.0o
3
87.4cos245o
87.4sin245o
R
and
Y = Rsin
Multiply and round to the least number of sig figs.
It should look like this:
Vector
X component
Y component
1
21.2
21.2
2
-47.9
36.1
3
-36.9
-79.2
R
Step 3:
Step 4:
Add the X components straight down to get the X component of the resultant. Do the same
with the Y components. When adding, choose the correct place value to add to.
Vector
X component
Y component
1
21.2
21.2
2
-47.9
36.1
3
-36.9
-79.2
R
-63.6
-21.9
Use the Pythagorean Theorem to find R.
R2 = (63.6)2 + (21.9)2 = 4040 + 480 = 4520
R = √4520 = 67.2 N
Step 5:
Use tan =
𝒀
𝑿
to find .
Tan =
−21.9
−63.6
= .344
 = tan-1(.344) = 19.0o
Since (-63.6,-21.9) should be in the 3rd quadrant, we add 180.000o to our angle.
Q = 19.0o + 180.0o = 199.0o
Final Answer:
R = 67.2 N, 199.0o
Tip to Tail Vector Addition (by hand)




You can add vectors by hand as well to give you a visual representation of a resultant vector
Given a set of vectors to add, choose one of them.
o Measure the vector out and draw it in an open area of the page
o Vectors must all be drawn proportionally (proportional lengths and angles)
Take ANY other vector in your set and place the tail end of it on the tip of the last vector (“tip to tail”
method)
After all vectors are added, use a straight edge to draw a line from the tail of the first vector to the tip
of the final vector.
EXAMPLE (draw the resultant vector):
Work Here
Now you try, but in a different order:
Work Here
