The displacement vector
is the vector sum of the jogger’s displacement vector
along the stairs (from point
A at the bottom of the stairs to point
T at the top of the stairs) and his displacement vector
on the top of the hill (from point
T at the top of the stairs to the fountain at point
B ). We must find the horizontal and the vertical components of
. If each step has width
w and height
h , the horizontal component of
must have a length of 200
w and the vertical component must have a length of 200
h. The actual distance the jogger covers is the sum of the distance he runs up the stairs and the distance of 50.0 m that he runs along the top of the hill.
Solution
In the coordinate system indicated in
[link] , the jogger’s displacement vector on the top of the hill is
. His net displacement vector is
Therefore, his displacement vector
along the stairs is
Its scalar components are
and
. Therefore, we must have
Hence, the step width is
w = 40.0 m/200 = 0.2 m = 20 cm, and the step height is
w = 30.0 m/200 = 0.15 m = 15 cm. The distance that the jogger covers along the stairs is
Thus, the actual distance he runs is
. When he makes a loop and comes back from the fountain to his initial position at point
A , the total distance he covers is twice this distance, or 200.0 m. However, his net displacement vector is zero, because when his final position is the same as his initial position, the scalar components of his net displacement vector are zero (
[link] ).
In many physical situations, we often need to know the direction of a vector. For example, we may want to know the direction of a magnetic field vector at some point or the direction of motion of an object. We have already said direction is given by a unit vector, which is a dimensionless entity—that is, it has no physical units associated with it. When the vector in question lies along one of the axes in a Cartesian system of coordinates, the answer is simple, because then its unit vector of direction is either parallel or antiparallel to the direction of the unit vector of an axis. For example, the direction of vector
is unit vector
. The general rule of finding the unit vector
of direction for any vector
is to divide it by its magnitude
V :
We see from this expression that the unit vector of direction is indeed dimensionless because the numerator and the denominator in
[link] have the same physical unit. In this way,
[link] allows us to express the unit vector of direction in terms of unit vectors of the axes. The following example illustrates this principle.
The unit vector of direction
If the velocity vector of the military convoy in
[link] is
, what is the unit vector of its direction of motion?
Strategy
The unit vector of the convoy’s direction of motion is the unit vector
that is parallel to the velocity vector. The unit vector is obtained by dividing a vector by its magnitude, in accordance with
[link] .
Solution
The magnitude of the vector
is
To obtain the unit vector
, divide
by its magnitude:
Significance
Note that when using the analytical method with a calculator, it is advisable to carry out your calculations to at least three decimal places and then round off the final answer to the required number of significant figures, which is the way we performed calculations in this example. If you round off your partial answer too early, you risk your final answer having a huge numerical error, and it may be far off from the exact answer or from a value measured in an experiment.
