Lesson 5
Adding and Subtracting
Vectors in a Plane
 Working with scaled vector diagramsWorking with scaled vector diagrams
(requires measuring length and using a(requires measuring length and using a
protractor)protractor)
 Understanding how to add and subtract non-Understanding how to add and subtract non-
linear vectors.linear vectors.
 Nelson reference pages:Nelson reference pages:
Page 58 – 64, 66- 68

Previously, we have worked with linearPreviously, we have worked with linear
vectors. These vectors had only twovectors. These vectors had only two
directions, positive (+) and negative (-)directions, positive (+) and negative (-)
When adding vectors that are not linear,When adding vectors that are not linear,
there are many different directions.there are many different directions.
 Example: Given;Example: Given; ΔdΔd11 = 4.0m [E], Δd= 4.0m [E], Δd22 = 3.0 m [N]= 3.0 m [N]
Determine ΔdDetermine Δd11 + Δd+ Δd22
To do this, we must first draw a vector diagram.To do this, we must first draw a vector diagram.

Adding Two Vectors
 Rules:Rules:
 Draw a compassDraw a compass
 The length of the vector arrowThe length of the vector arrow
must represent the magnitudemust represent the magnitude
 To add: Place the tail of thePlace the tail of the
second vector at the tip of thesecond vector at the tip of the
first vector. Thefirst vector. The resultant ((RR))
is a vector drawn from the tailis a vector drawn from the tail
of the first to the tip of theof the first to the tip of the
second.second.
 Vectors being added mustVectors being added must
have the same unit ofhave the same unit of
measure – here it is “m”measure – here it is “m”
N
W E
A = 4 . 0 m
B=3.0m
D e t 'm : A + B
N
W E
A = 4 . 0 m
B=3.0m
R
=
A
+B

Subtracting Two VectorsSubtracting Two Vectors
To subtract two vectors:To subtract two vectors:
 We just draw theWe just draw the
vector beingvector being
subtracted in thesubtracted in the
opposite direction andopposite direction and
then follow the rulesthen follow the rules
for vector addition.for vector addition.
 A – B = A + (-B)A – B = A + (-B)
N
W E
A = 4 . 0 m
B=3.0m
D e t 'm : A - B
N
W E
A = 4 .0 m
-B=3.0m
R
=
A
- B

Finding the Resultant
 The resultant can be found two ways:The resultant can be found two ways:
 The first method is to use aThe first method is to use a scaled vector
diagram. This method will be required if. This method will be required if
working with more than two vectors.working with more than two vectors.
 A chalkboard example will be used.
 Students should refer to Sample Problems 1 & 2 on
pages 62-63
 The second method is toThe second method is to use trigonometry..
We will always use this method when workingWe will always use this method when working
with justwith just two vectors since a triangle is formedvectors since a triangle is formed
when the resultant vector is drawn in.when the resultant vector is drawn in.

Using Trig to Find RUsing Trig to Find R
 The first step is always toThe first step is always to
draw the vector diagram.draw the vector diagram.
 Using Pythagorean thm weUsing Pythagorean thm we
can find the magnitude ofcan find the magnitude of RR
to be 5.0 m. The directionto be 5.0 m. The direction
is measured from the tail ofis measured from the tail of
RR.. θθ == TanTan-1-1
(3/4) =37(3/4) =3700
 If a right angle triangle isIf a right angle triangle is
not present, cosine law maynot present, cosine law may
be required to get thebe required to get the
angle.angle.
 R = 5.0m [E37R = 5.0m [E3700
N]N]
N
W E
A = 4 .0 m
B=3.0m
R =
A
+B
θ

Other Points
 WhenWhen writing the
direction of a vector,of a vector,
choose a primarychoose a primary
direction (N, S, E, W,direction (N, S, E, W,
Up, Down). ThenUp, Down). Then
determine the angle anddetermine the angle and
direction for thedirection for the
deviation from thedeviation from the
primary direction.primary direction.
 Write directions withWrite directions with
angles less than 90angles less than 9000
 There are two possible
answers – as shown.
4 0 0
U p
d
E a s t
d = 5 .0 m [ E 4 0 0 U p ]
d = 5 .0 m [ U p 5 0 0
E ]

Practice Questions
Nelson Textbook:
 Page 65, # 1-3, forPage 65, # 1-3, for 3b do not recopy the diagram;do not recopy the diagram;
just measure from the TB.just measure from the TB.
 Page 65, # 4, 5, 7, draw the vector diagram (notPage 65, # 4, 5, 7, draw the vector diagram (not
to scale) for both questions and use trigonometryto scale) for both questions and use trigonometry
to solve.to solve.
Questions from McGraw-Hill TB::
1.1. A boat heads from port, in still water, and travelsA boat heads from port, in still water, and travels
north 21.0km. It then travels 30.0 km [W 30 S]. Itnorth 21.0km. It then travels 30.0 km [W 30 S]. It
finally heads 36.0 km [W10N]. a.) Determinefinally heads 36.0 km [W10N]. a.) Determine ΔdΔdTT
b.) The direction the boat should head to returnb.) The direction the boat should head to return
directly home from its final position.directly home from its final position.
Ans 62.6 km [W11N], [E11S]

1.1. A hockey puck hits the boards at 12 m/sA hockey puck hits the boards at 12 m/s
at an angle of 30 degrees to the boards.at an angle of 30 degrees to the boards.
It is then deflected with at 10 m/s at anIt is then deflected with at 10 m/s at an
angle of 25 degrees to the boards.angle of 25 degrees to the boards.
Determine the pucks change in velocity.Determine the pucks change in velocity.
(Consider the boards as extending in an(Consider the boards as extending in an
W-E direction and the puck starting itsW-E direction and the puck starting its
motion from the west side and movingmotion from the west side and moving
north it then deflects in a south easterlynorth it then deflects in a south easterly
direction)direction)
Ans 10 m/s [70
to the normal from boards]

1.1. A hockey puck hits the boards at 12 m/sA hockey puck hits the boards at 12 m/s
at an angle of 30 degrees to the boards.at an angle of 30 degrees to the boards.
It is then deflected with at 10 m/s at anIt is then deflected with at 10 m/s at an
angle of 25 degrees to the boards.angle of 25 degrees to the boards.
Determine the pucks change in velocity.Determine the pucks change in velocity.
(Consider the boards as extending in an(Consider the boards as extending in an
W-E direction and the puck starting itsW-E direction and the puck starting its
motion from the west side and movingmotion from the west side and moving
north it then deflects in a south easterlynorth it then deflects in a south easterly
direction)direction)
Ans 10 m/s [70
to the normal from boards]