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] .
t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080
Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?