ADDING
VECTORS
•
Many
stops and starts over the course of the day result in many changes in
position. As you walk around the
hallways at school, your travels could be represented by a series of
displacement vectors.
•
To
begin, vectors in one dimension will be
considered, either in the same direction or opposite directions.
•
There
are three methods for solving these vector addition problems:
1.
Scale Vector Diagrams
a.
Must
consider both the size and direction of each quantity being added.
b.
Add
vectors using “head-to-tail” rule – Join each vector by connecting the “head”
end of one vector to the “tail” end of the next vector.
c.
Find the resultant (the final sum of the vectors)
by drawing an arrow from the tail of the first vector to the head of the last
vector. You‘ve found the resultant displacement (ΔdR)!
d.
This
process is very efficient when dealing with vectors that are not on a straight
line.
e.
Summary
of process:
i.
State
the directions (ex. with a compass symbol like [N], etc.)
ii.
List
the givens and indicate what variable is being solved.
iii.
State
the scale to be used (ex. 1 cm = 5 km)
iv.
Draw
one of the initial vectors exactly to scale.
v.
Join
the second and additional vectors head to tail and to scale.
vi.
Draw
and label the resultant vector.
vii.
Measure
the resultant vector and convert the length using your scale.
viii.
Write
your answer including both the size and direction of the resultant vector.
2.
Adding Vectors
Algebraically
a.
Add
vectors by assigning a positive or negative direction to the value of the
quantity.
b.
State
the convention used in each question.
Most often north and east are stated as the positive directions.
c.
At
the end of a problem, the positive or negative sign need to be converted back
into a direction.
d.
Summary
of process:
i.
Indicate
which direction is positive and which is negative.
ii.
List
the givens and indicate what variable is being solved.
iii.
Write
the equation for adding vectors.
iv.
Substitute
numbers (with correct signs) into the equation and solve.
v.
Write
your answer including both the size and direction.
3.
A Combined Method – a combination of the
above two methods.
a.
The
inefficiency of drawing scale diagrams is removed but the visual advantage of
vector diagrams is retained.
b.
Vectors
can just be sketched approximately accurately, relative to each other, and the
vector addition is done algebraically.
c.
Summary
of process:
i.
State
which direction is positive and which is negative.
ii.
Sketch
a labeled vector diagram – not to scale, but using relative sizes.
iii.
Write
the equation for adding the vectors.
iv.
Substitute
numbers (with correct signs) into the equation, and solve.
v.
Write
your answer including both the size and direction.