Let us return to the right triangle in
[link] . The quotient of the adjacent side
to the hypotenuse
A is the cosine function of direction angle
,
, and the quotient of the opposite side
to the hypotenuse
A is the sine function of
,
. When magnitude
A and direction
are known, we can solve these relations for the scalar components:
When calculating vector components with
[link] , care must be taken with the angle. The direction angle
of a vector is the angle measured
counterclockwise from the positive direction on the
x -axis to the vector. The clockwise measurement gives a negative angle.
Components of displacement vectors
A rescue party for a missing child follows a search dog named Trooper. Trooper wanders a lot and makes many trial sniffs along many different paths. Trooper eventually finds the child and the story has a happy ending, but his displacements on various legs seem to be truly convoluted. On one of the legs he walks 200.0 m southeast, then he runs north some 300.0 m. On the third leg, he examines the scents carefully for 50.0 m in the direction
west of north. On the fourth leg, Trooper goes directly south for 80.0 m, picks up a fresh scent and turns
west of south for 150.0 m. Find the scalar components of Trooper’s displacement vectors and his displacement vectors in vector component form for each leg.
Strategy
Let’s adopt a rectangular coordinate system with the positive
x -axis in the direction of geographic east, with the positive
y -direction pointed to geographic north. Explicitly, the unit vector
of the
x -axis points east and the unit vector
of the
y -axis points north. Trooper makes five legs, so there are five displacement vectors. We start by identifying their magnitudes and direction angles, then we use
[link] to find the scalar components of the displacements and
[link] for the displacement vectors.
Solution
On the first leg, the displacement magnitude is
and the direction is southeast. For direction angle
we can take either
measured clockwise from the east direction or
measured counterclockwise from the east direction. With the first choice,
. With the second choice,
. We can use either one of these two angles. The components are
The displacement vector of the first leg is
On the second leg of Trooper’s wanderings, the magnitude of the displacement is
and the direction is north. The direction angle is
. We obtain the following results:
On the third leg, the displacement magnitude is
and the direction is
west of north. The direction angle measured counterclockwise from the eastern direction is
. This gives the following answers:
On the fourth leg of the excursion, the displacement magnitude is
and the direction is south. The direction angle can be taken as either
or
. We obtain
