Lesson 2.2
Vector Addition Through Graphical Method
Vectors in One Dimension
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If two or more vectors are pointing in the
same direction, it means that they are
parallel to each other, as shown in Fig. 2.2.2.
When two vectors have opposite directions
but have the same magnitude, they are
called antiparallel.
*Notice that the
length of is similar to
the length of , but it is
pointing in the
opposite direction.
Addition of Vectors
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To get the total displacement, we only need
to consider the initial and the starting points.
The initial point of the total displacement is
the starting point of A while the endpoint is
the endpoint of vector B.
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We call this displacement as R , also called
the vector sum or resultant. It can be
How are vectors added graphically?
#1 - Suppose a person covered two different
displacements: A = 30 m, 25° north of east and B =
55 m, 70° south of east. What is her total
displacement?
1. Draw vector on a graphing or bond paper.
Make sure that the magnitude is represented
by a proper scale.
expressed mathematically as R = A + B.
Head to tail method, it is where the tail of
the second vector is drawn at the head or
tip of the first vector.
Graphical Method of Adding Vectors
2. Draw vector B using the same scale that you
used1 in the previous vector. Its tail should
start from the tip of A.
Lesson 2.2
Vector Addition Through Graphical Method
3. Draw an arrow connecting the tail of A and the
tip of B . This is the resultant vector R .
4. Measure the length of and use the scale to find
its real length. Use the protractor to measure the
angle.
5. If there is another vector, simply connect it to the
tail of the previous vector and follow similar steps.
Let's Practice!
*Scale: 1cm = 20m
*Scale: 1cm = 1 000m
Lesson 2.3
Components of Vectors
Even though the graphical method of adding vectors
is effective in visualizing the process, its final answer
is not that accurate due to the precision of the
measuring devices used. This is the reason why the
analytical method is used to get answers more
accurately. Let us first discuss a topic essential in
the analytical method of adding vectors—
components of vectors.
Components of a Vector
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A vector directed at an angle from the
horizontal or vertical axis can be resolved
into its components.
Consider a displacement vector pointing in
the northeast direction.
It can be divided into its two components: a
component along the east (horizontal axis)
and another component along the north
(vertical axis), as shown in Fig. 2.3.2; also
composed of a component along the south
and another component along the west
direction.
The components can be calculated if the
magnitude and the direction of the vector are
given. However, it is important to specify first
the reference direction.
The direction of the vector can be presented
as an angle 𝜃 (Greek letter for theta),
measured in a counterclockwise direction
from the +x-axis.
calculated using the trigonometric functions
sine, cosine, and tangent.
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The x-component of vector A is given below.
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The y-component of vector A is given below.
*In both equations, A is the magnitude of the
vector and 𝜃 is the angle as measured from the
positive x-axis.
Let's Practice!
Let's Practice!
Calculating the Resultant Vector
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A vector’s magnitude and direction can also be
calculated using its components.
The magnitude of vector can be calculated
using the equation below based on the
Pythagorean theorem.
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The angle or direction is calculated using the
inverse function of tangent as shown in the
equation below.
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To ensure that your final answer is correct it is
very important that you check the signs of the xand y-components.
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After identifying the location, check the
sign of the angle 𝜃 you calculated.
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A positive sign means that you have to measure
in a counterclockwise direction from one of the
axes;
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While a negative sign indicates that you have to
measure the angle in a clockwise direction.