Name: ______________________________
Physics
Date: _________
1.3 Algebraically Adding Vectors
Algebraically Adding Vectors:
When vectors are parallel to each other, it is easy to just add or subtract them.
3m
+
=
5m
3m
5m
8m
When vectors are at perpendicular (right angles) to each other, it is possible to algebraically (mathematically) add the
vectors using trigonometry.
+
3m
4m
Step 1:
Draw the vectors so that they are connected head to tail.
If you skip this step the direction will be incorrect.
Step2:
Draw in the resultant vector to make a right triangle.
3m
3m
4m
4m
Step 3:
Find the Magnitude using Pythagoreans theorem
2
2
a +b = c
Step 4:
Find the direction using the tangent function.
tan θ =
2
c = √a2 + b 2
opposite
adjacent
θ = tan−1
opposite
adjacent
c
b
3m
θ
a
4m
Directions: Find the resultant of the following vectors algebraically by adding them head to tail (redraw one of the
vectors onto another). The magnitudes of each vector are given, but they are not drawn to scale.
1.
3
5
2.
10
25
3. Connect the vectors head to tail to form a right triangle, and then find the resultant.
4
3
6
The Parallelogram Rule:
The two vectors on the right act on an object.
What is the resultant vector?
The Parallelogram Rule helps combine
these vectors into the resultant.
(Note: The tails of the vectors must be connected to use this rule. If they are not connected,
you must redraw them so that they are connected before completing the steps below.)
Step 1
Step 2
Step 3
Redraw one of the vectors.
Redraw the second vector to
Draw in the resultant vector.
Choose a horizontal or vertical
make a parallelogram.
The tail should start where all
vector if possible because you
the tails meet, then go
know what angle to draw it with.
diagonally across.
In this case we drew the
horizontal vector. Measure the
magnitude with a ruler, then
draw it.
Directions: Use the parallelogram rule to carefully construct the resultants for the vectors below. Use a ruler to draw
vectors to scale.
4.
5.
6.
7.
The Head to Tail Method:
The two vectors on the right act on an object.
What is the resultant vector?
The Head to Tail Method helps combine
these vectors into the resultant.
Step 1
Redraw one of the vectors by connecting the head of one to the tail
of the other.
(Choose a horizontal or vertical vector if possible because you know
what angle to draw it with)
In this case we drew the horizontal vector.
(Measure the magnitude with a ruler)
Step 2
Draw in the resultant vector.
The tail should start at the open
tail of the vector and end at the
last head that you drew.
Directions: Use the head to tail method to find the resultants for the vectors below.
8.
9.
10. Answer the following questions based on the diagram to the right.
a. Using a ruler, determine the scale used in the diagram.
1 cm = ___________ units
b. Draw the resultant force vector to scale on the diagram.
Label the vector R.
c. Determine the magnitude of the resultant vector.
25
R = ____________ units
d. Determine the measure of the angle, in degrees, between
north and the resultant vector, R.
15
θ = ____________ o