Answer to Essential Question 1.4: Other examples of scalars include mass, distance, and speed.
Examples of vectors, which have directions associated with them, include displacement, force,
and acceleration.
1-5 Adding Vectors
EXAMPLE 1.5 – Adding vectors
Let’s define a vector as being the sum of the two vectors and from Exploration
1.4. A vector that results from the addition of two or more vectors is called a resultant vector.
(a) Draw the vectors and tip-to-tail to show geometrically the resultant vector .
(b) Use the components of vectors and to find the components of .
(c) Express in unit-vector notation.
(d) Express in terms of its magnitude and direction.
SOLUTION
(a) To add the vectors geometrically
we can move the tail of to the tip of ,
or the tail of to the tip of . The order
makes no difference. If we had more
vectors, we could continue the process,
drawing them tip-to-tail in sequence. The
resultant vector always goes from the tail of
the first vector to the tip of the last vector,
as is shown in Figure 1.7.
(b) Now let’s add the vectors using
their components. We already know the x
and y components of and (see
Exploration 1.4), so we can use those to find
the components of the resultant vector .
Table 1.2 demonstrates the process. Note
that the components of are
shown here to two decimal
places, even though we know
them with more precision.
Because we’ll be adding the
components of to the
components of , which we
know to two decimal places,
our final answers should also
be expressed with two
decimal places.
Chapter 1 – Introduction to Physics Page 1 – 10
Vector
x-component
y-component
Figure 1.7: Adding vectors geometrically, tip-to-tail. In
(a), the tail of vector is placed at the tip of ; in (b), the
tail of vector is placed at the tip of . The same resultant
vector is produced - the order does not matter.
Table 1.2: Adding the vectors and using components. The process is
shown pictorially in Figure 1.8.
