PHYSICS 30: Vectors – Symbols & Directions
Vector: a measurement with two parts: a magnitude (the measurement)
and a direction.
The following measurements are vector quantities:

Position:
d
4 km [North] of the city

Displacement:
Δd
275 m [S]

Velocity:
v
80 km/h [W]

Acceleration:
a
9.8 m/s2 [Down]

Force:
F
250 Newtons [North] = 250 N [N]

Momentum:
p
120 (kg●m)/s [Right]
A vector can be drawn as a line with an arrowhead (ray).
 The length of the ray represents the magnitude of the vector.
 The arrowhead shows the direction of the vector.
A vector diagram is a picture of some math statement.
Vectors can be added “Tip-to-Tail” (Like Elephants!)
20 metres [North] + 30 metres [East] = Δd
20 m [N] + 30 m [E] = Δd
30 m [E]
20 m [N]
Δd
Use Pythagoras and SOHCAHTOA to help solve right-angle triangles.
a2 + b2 = c2
sin θ = opp/hyp
cos θ = adj/hyp
tan θ = opp/adj
B
hypotenuse
c
θ
opposite
a
90º
b adjacent
C
A
Use the Law of Sines and the Law of Cosines to solve triangles that
don’t have a right angle.
C
b
A
a
c
Law of Sines:
Sin A = Sin B = Sin C
a
b
B
c
Law of Cosines:
a2 = b2 + c2 – 2bc CosA
b2 = a2 + c2 – 2ac CosB
c2 = a2 + b2 – 2ab CosC
Direction: the direction of a vector can be expressed using north, east,
west, or south as a reference direction.
N
W
E
20º
[E 20º S]
70º
[S 70º E]
or 110º
(measured clockwise from North)
Heading & Bearing:
Objects like planes and boats can be affected by outside forces such as wind and
S
waves. There can be a difference in the direction an object is pointed and the
direction it actually travels.
WINDY
HEADING: the direction the plane
is pointed/aimed/headed
Object + Effect = Result
Heading + Effect = Bearing
Examples:
plane + wind = result
or
boat + current = result
BEARING: the
direction the plane
actually travels
PHYSICS 30: Vectors – Practice
1. Use the three vectors to draw and calculate each math statement.
A = 10 m [Right]
a. A + B = Δd
b. B + C = Δd
c. A + C = Δd
d. B + A = Δd
B = 6 m [Up]
C = 4 m [Left]
2. Draw in and calculate the north, south, east, or west components of each vector.
a. Displacement
400 km [S 40º E]
b. Velocity
50 m/s [R 60° Up]
c. Acceleration
d. Force
6.8 m/s2 [L 15° Dn]
1200 Newtons [W 40° N] = 1200 N [W 40º N]
PHYSICS 30: Vectors – Components of Vectors – Problem Practice
1. A glider flies 15 km [W]. The pilot turns the glider to fly for 12 km in the direction
[N 40º W].
a. Write a math statement for the displacement of the glider.
b. Draw a vector diagram for this math statement. Don’t solve the triangle.
c. Find the components of the 12 km [N 40º W] vector.
d. Rewrite and simplify the math statement for the glider’s displacement.
e. Draw a vector diagram for the glider’s displacement.
f. Solve the vector diagram to determine the magnitude and direction of the
glider’s displacement.
2. A sailboat sails for 5 km in the direction [S 70º E]. The sailboat then turns and sails
for 8 km in the direction [W 30º S].
a. Write a math statement for the displacement of the sailboat.
b. Draw a vector diagram for the sailboat’s displacement. Don’t solve the triangle.
c. Find the components for the two vectors.
d. Rewrite and simplify the math statement for the sailboat’s displacement.
e. Draw a vector diagram for the sailboat’s displacement.
f. Solve the vector diagram to determine how far and in what direction the
sailboat is from where it started.
PHYSICS 30: Vectors – Heading & Bearing Practice Questions
1. A river has a current that flows from west to east. You want to travel directly across
the river from the south bank to the north. Because of your knowledge of physics you
decide to head your boat :
i. Due north
ii. Due west
iii. In a northwesterly direction
iv. In a northeasterly direction
b. Explain your choice.
2. The same river flows towards the east. Because of your knowledge of physics you
head your boat in the direction of [N 53° W]. You end up traveling with a resulting
velocity of 6.0 m/s [N].
a. What is the velocity of the current?
b. What is the boat speed relative to the water? (This is the speed the boat
would be capable of in still water.)
3. A boat is capable of traveling at a speed of 12 m/s relative to the water (the speed of
the boat without a current). The driver of the boat wishes to cross directly from the
south shore to the north shore at a point where the river is 800 metres wide. The
current is flowing [East] at a rate of 2.5 m/s.
a. Write a math statement representing the combining of the boat and current to
produce a resulting trip directly across the river.
b. Draw a vector diagram for your math statement.
c. Calculate the heading the driver of the boat must steer.
d. Calculate the resulting velocity of the boat.
e. Calculate how long it will take the boat to cross the river.
4. A pilot takes off from Regina to travel to Winnipeg which is 600 km [E] of Regina.
There is considerable cloud cover so the pilot maintains a speed and heading of 150
km/h [East] using his air speed indicator and compass. The pilot is unaware of a wind
blowing the plane south at a rate of 45 km/h (he doesn’t know the air is moving).
a. Write a math statement representing the plane’s heading combined with the
wind.
b. Draw a vector diagram for the math statement.
c. Does the plane fly towards Winnipeg? What direction does it end up traveling?
d. What direction should the pilot head the plane if he wants to fly directly to
Winnipeg?
e. What will be the resulting velocity of the plane if the pilot steers the correct
heading so the plane flies directly to Winnipeg?
f. How long will it take the plane to travel from Regina to Winnipeg?
5. A small plane is headed south with a relative velocity of 160 km/h to the air (this is
its air speed). After two hours the pilot spots a landmark and checking with a map
determines that the plane has traveled only 420 km from its starting point. Also from
the map the pilot determines that the plane has traveled in a southeasterly direction.
a. Calculate the wind speed if the direction of the wind is [East].