We now have the tools to solve the problem we introduced in the opening of the section.
An airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of 140°. A north wind (from north to south) is blowing at 16.2 miles per hour. What are the ground speed and actual bearing of the plane? See
[link] .
The ground speed is represented by
in the diagram, and we need to find the angle
in order to calculate the adjusted bearing, which will be
Notice in
[link] , that angle
must be equal to angle
by the rule of alternating interior angles, so angle
is 140°. We can find
by the Law of Cosines:
The ground speed is approximately 213 miles per hour. Now we can calculate the bearing using the Law of Sines.
Therefore, the plane has a SE bearing of 140°+2.8°=142.8°. The ground speed is 212.7 miles per hour.
The position vector has its initial point at the origin. See
[link] .
If the position vector is the same for two vectors, they are equal. See
[link] .
Vectors are defined by their magnitude and direction. See
[link] .
If two vectors have the same magnitude and direction, they are equal. See
[link] .
Vector addition and subtraction result in a new vector found by adding or subtracting corresponding elements. See
[link] .
Scalar multiplication is multiplying a vector by a constant. Only the magnitude changes; the direction stays the same. See
[link] and
[link] .
Vectors are comprised of two components: the horizontal component along the positive
x -axis, and the vertical component along the positive
y -axis. See
[link] .
The unit vector in the same direction of any nonzero vector is found by dividing the vector by its magnitude.
The magnitude of a vector in the rectangular coordinate system is
See
[link].
In the rectangular coordinate system, unit vectors may be represented in terms of
and
where
represents the horizontal component and
represents the vertical component. Then,
v = a
i + b
j is a scalar multiple of
by real numbers
See
[link] and
[link] .
Adding and subtracting vectors in terms of
i and
j consists of adding or subtracting corresponding coefficients of
i and corresponding coefficients of
j . See
[link] .
A vector
v =
a
i +
b
j is written in terms of magnitude and direction as
See
[link] .
The dot product of two vectors is the product of the
terms plus the product of the
terms. See
[link] .
We can use the dot product to find the angle between two vectors.
[link] and
[link] .
Dot products are useful for many types of physics applications. See
[link] .
the transfer of energy by a force that causes an object to be displaced; the product of the component of the force in the direction of the displacement and the magnitude of the displacement
A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.
A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike
Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.
the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon