The three displacement vectors
,
, and
in
[link] are specified by their magnitudes
A = 10.0,
B = 7.0, and
C = 8.0, respectively, and by their respective direction angles with the horizontal direction
,
, and
. The physical units of the magnitudes are centimeters. Choose a convenient scale and use a ruler and a protractor to find the following vector sums: (a)
, (b)
(c)
.
Vectors used in
[link] and in the Check Your Understanding feature that follows.
Strategy
In geometric construction, to find a vector means to find its magnitude and its direction angle with the horizontal direction. The strategy is to draw to scale the vectors that appear on the right-hand side of the equation and construct the resultant vector. Then, use a ruler and a protractor to read the magnitude of the resultant and the direction angle. For parts (a) and (b) we use the parallelogram rule. For (c) we use the tail-to-head method.
Solution
For parts (a) and (b), we attach the origin of vector
to the origin of vector
, as shown in
[link] , and construct a parallelogram. The shorter diagonal of this parallelogram is the sum
. The longer of the diagonals is the difference
. We use a ruler to measure the lengths of the diagonals, and a protractor to measure the angles with the horizontal. For the resultant
, we obtain
R = 5.8 cm and
. For the difference
, we obtain
D = 16.2 cm and
, which are shown in
[link] .
Using the parallelogram rule to solve (a) (finding the resultant, red) and (b) (finding the difference, blue).
For (c), we can start with vector
and draw the remaining vectors tail-to-head as shown in
[link] . In vector addition, the order in which we draw the vectors is unimportant, but drawing the vectors to scale is very important. Next, we draw vector
from the origin of the first vector to the end of the last vector and place the arrowhead at the end of
. We use a ruler to measure the length of
, and find that its magnitude is
S = 36.9 cm. We use a protractor and find that its direction angle is
. This solution is shown in
[link] .
Using the tail-to-head method to solve (c) (finding vector
, green).
Check Your Understanding Using the three displacement vectors
,
, and
in
[link] , choose a convenient scale, and use a ruler and a protractor to find vector
given by the vector equation
.
A vector quantity is any quantity that has magnitude and direction, such as displacement or velocity. Vector quantities are represented by mathematical objects called vectors.
Geometrically, vectors are represented by arrows, with the end marked by an arrowhead. The length of the vector is its magnitude, which is a positive scalar. On a plane, the direction of a vector is given by the angle the vector makes with a reference direction, often an angle with the horizontal. The direction angle of a vector is a scalar.
Two vectors are equal if and only if they have the same magnitudes and directions. Parallel vectors have the same direction angles but may have different magnitudes. Antiparallel vectors have direction angles that differ by
. Orthogonal vectors have direction angles that differ by
.
When a vector is multiplied by a scalar, the result is another vector of a different length than the length of the original vector. Multiplication by a positive scalar does not change the original direction; only the magnitude is affected. Multiplication by a negative scalar reverses the original direction. The resulting vector is antiparallel to the original vector. Multiplication by a scalar is distributive. Vectors can be divided by nonzero scalars but cannot be divided by vectors.
Two or more vectors can be added to form another vector. The vector sum is called the resultant vector. We can add vectors to vectors or scalars to scalars, but we cannot add scalars to vectors. Vector addition is commutative and associative.
To construct a resultant vector of two vectors in a plane geometrically, we use the parallelogram rule. To construct a resultant vector of many vectors in a plane geometrically, we use the tail-to-head method.
