Understand the rules of vector addition, subtraction, and multiplication.
Apply graphical methods of vector addition and subtraction to determine the displacement of moving objects.
Displacement can be determined graphically using a scale map, such as this one of the Hawaiian Islands. A journey from Hawai’i to Moloka’i has a number of legs, or journey segments. These segments can be added graphically with a ruler to determine the total two-dimensional displacement of the journey. (credit: US Geological Survey)
Vectors in two dimensions
A
vector is a quantity that has magnitude and direction. Displacement, velocity, acceleration, and force, for example, are all vectors. In one-dimensional, or straight-line, motion, the direction of a vector can be given simply by a plus or minus sign. In two dimensions (2-d), however, we specify the direction of a vector relative to some reference frame (i.e., coordinate system), using an arrow having length proportional to the vector’s magnitude and pointing in the direction of the vector.
[link] shows such a
graphical representation of a vector , using as an example the total displacement for the person walking in a city considered in
Kinematics in Two Dimensions: An Introduction . We shall use the notation that a boldface symbol, such as
, stands for a vector. Its magnitude is represented by the symbol in italics,
, and its direction by
.
Vectors in this text
In this text, we will represent a vector with a boldface variable. For example, we will represent the quantity force with the vector
, which has both magnitude and direction. The magnitude of the vector will be represented by a variable in italics, such as
, and the direction of the variable will be given by an angle
.
A person walks 9 blocks east and 5 blocks north. The displacement is 10.3 blocks at an angle
north of east.To describe the resultant vector for the person walking in a city considered in
[link] graphically, draw an arrow to represent the total displacement vector
. Using a protractor, draw a line at an angle
relative to the east-west axis. The length
of the arrow is proportional to the vector’s magnitude and is measured along the line with a ruler. In this example, the magnitude
of the vector is 10.3 units, and the direction
is
north of east.
Vector addition: head-to-tail method
The
head-to-tail method is a graphical way to add vectors, described in
[link] below and in the steps following. The
tail of the vector is the starting point of the vector, and the
head (or tip) of a vector is the final, pointed end of the arrow.
Head-to-Tail Method: The head-to-tail method of graphically adding vectors is illustrated for the two displacements of the person walking in a city considered in
[link] . (a) Draw a vector representing the displacement to the east. (b) Draw a vector representing the displacement to the north. The tail of this vector should originate from the head of the first, east-pointing vector. (c) Draw a line from the tail of the east-pointing vector to the head of the north-pointing vector to form the sum or
resultant vector
. The length of the arrow
is proportional to the vector’s magnitude and is measured to be 10.3 units . Its direction, described as the angle with respect to the east (or horizontal axis)
is measured with a protractor to be
.
