40S Applied Math
Mr. Knight – Killarney School
Unit: Vectors
Lesson: VEC-L2
Vector Addition &
Scalar Multiplication
Resultant Vector
Component
Vector 2
Component
Vector 1
VEC-L2 Objectives:
Add Vectors (Triangle & Parallelogram Methods)
Perform Scalar Multiplication on Vectors
Learning Outcome B-2
Slide 1
40S Applied Math
Mr. Knight – Killarney School
Unit: Vectors
Lesson: VEC-L2
Whenever two vectors act on an object, the overall
effect of the vectors can be found by adding them. In
vector addition, we place the tail of the second
vector at the head of the first vector. The sum of the
vectors, called the resultant vector, is the vector that
goes from the tail of the first vector to the head of the
second one.
Theory – Vector Addition
Slide 2
40S Applied Math
Mr. Knight – Killarney School
Unit: Vectors
Lesson: VEC-L2
An athlete is swimming downstream in a river with a
current of 0.5 m/s. The swimmer's speed in still water
would be 1.2 m/s. What is the swimmer's speed with
respect to the shore?
When adding vectors, we place the tail of the second
vector at the head of the first. The resultant vector
represents the velocity of the swimmer as observed by
someone on shore.
Theory – Addition, Same
Direction
Slide 3
40S Applied Math
Mr. Knight – Killarney School
Unit: Vectors
Lesson: VEC-L2
The athlete from the previous page is now swimming in the
opposite direction against the current. The vectors representing
the swimmer and the current are shown below.
We again add the vectors by placing the tail of the second vector
at the head of the first. The resultant vector represents the
velocity of the swimmer as observed by someone on shore.
Theory – Addition, Opposite
Direction
Slide 4
40S Applied Math
Mr. Knight – Killarney School
Unit: Vectors
Lesson: VEC-L2
When we multiply a vector by a scalar, the magnitude of the
vector is multiplied by the scalar and the direction does not
change.
Mike's turtle walks 12 feet east per minute towards the river.
Draw vectors that represent the turtle's displacement in one
minute and in three minutes.
In one minute:
In three minutes:
Theory – Scalar Multiplication
Slide 5
40S Applied Math
Mr. Knight – Killarney School
Unit: Vectors
Lesson: VEC-L2
Vectors aren’t always collinear.
When we solve problems that involve vectors, we usually draw a sketch
of the situation. This sketch will always have at least three vectors:
•two or more component vectors, which are the vectors that make up
the input to the problem and are added together to get the answer. The
vectors are component vectors.
•a resultant vector, which is the result of adding the components. This
is usually the answer to the vector problem.
Triangle Law of Addition:
We again add the vectors by
placing the tail of the second
vector at the head of the first. The
resultant vector represents the
velocity of the swimmer as
observed by someone on shore.
Theory – Triangle Law of
Addition
Slide 6
40S Applied Math
Mr. Knight – Killarney School
Unit: Vectors
Lesson: VEC-L2
Use the vectors shown to the following calculations. Just sketch
rough answers; you do not need to draw scale diagrams.
Examples for Practice
Slide 7
40S Applied Math
Mr. Knight – Killarney School
Unit: Vectors
Lesson: VEC-L2
A swimmer heads north across the river from one shore to the
other at 1.2 m/s. The river current is 0.5 m/s east. Determine the
magnitude and direction of the resultant velocity of the swimmer.
The vectors S and R represent the velocities of the swimmer in
still water and the river respectively.
Theory – Triangle Addition
Example
Slide 8
40S Applied Math
Mr. Knight – Killarney School
Unit: Vectors
Lesson: VEC-L2
To determine the magnitude and direction of R , we need to draw
a scale diagram of the triangle using pencil, paper, ruler, and
protractor. A suitable scale might be: 1 cm = 0.1 m/sec.
We now measure the length of the
resultant using the ruler, and the
direction of the resultant using a
protractor. The resultant vector
has a length of 13 cm, which
represents a magnitude of 1.3
m/sec, and a direction of N23°E.
Theory – Triangle Addition
Example
Slide 9
40S Applied Math
Mr. Knight – Killarney School
Unit: Vectors
Lesson: VEC-L2
A plane flies 240 km south, then 140 km west. The pilot wants to
return directly to the starting point. How far and in what direction
must he fly?
Examples for Practice
Slide 10
40S Applied Math
Mr. Knight – Killarney School
Unit: Vectors
Lesson: VEC-L2
A plane flies 240 km south, then 140 km west. The pilot wants to
return directly to the starting point. How far and in what direction
must he fly?
Examples for Practice
Slide 11
40S Applied Math
Mr. Knight – Killarney School
Unit: Vectors
Lesson: VEC-L2
An ultralight plane is headed N30ºW at 40 km/h. A 12 km/h wind
is blowing in the direction E20ºS. What is the resultant velocity of
the ultralight plane with respect to the ground?
Examples for Practice
Slide 12
40S Applied Math
Mr. Knight – Killarney School
Unit: Vectors
Lesson: VEC-L2
An ultralight plane is headed N30ºW at 40 km/h. A 12 km/h wind
is blowing in the direction E20ºS. What is the resultant velocity of
the ultralight plane with respect to the ground?
Examples for Practice
Slide 13
40S Applied Math
Mr. Knight – Killarney School
Unit: Vectors
Lesson: VEC-L2
In the previous examples, we used the triangle method of vector
addition, where we attach the tail of the second vector to the
head of the first, and then draw the resultant from the tail of the
first to the head of the second to complete a triangle. We may
also use the parallelogram method illustrated below.
Component vectors a & b represent two forces acting on an
object.
In this case, we arrange the
component vectors tail to tail, and
complete a parallelogram as shown by
the red arrows. The resultant is the
diagonal of the parallelogram.
Theory – Parallelogram Method
Slide 14
40S Applied Math
Mr. Knight – Killarney School
Unit: Vectors
Lesson: VEC-L2
Kathy and Brenda pull a wagon by exerting forces of 36 N and 42
N respectively using ropes attached to the front of the wagon.
The angle between the ropes is 30º. What is the resultant force
on the wagon?
Set your scale, then solve using the parallelogram method.
Examples for Practice
Slide 15
40S Applied Math
Mr. Knight – Killarney School
Unit: Vectors
Lesson: VEC-L2
Kathy and Brenda pull a wagon by exerting forces of 36 N and 42
N respectively using ropes attached to the front of the wagon.
The angle between the ropes is 30º. What is the resultant force
on the wagon?
Examples for Practice
Slide 16