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] .
Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.
Rafael
vjuvu
Elgoog
what is big bang theory?
Rosemary
what type of activity astronomer do?
Rosemary
No
Richard
the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen
according to the theory of astronomers why the moon is always appear in an elliptical orbit?
Gatjuol
hi !!! I am new in astronomy....
I have so many questions in mind ....
all of scientists of the word they just give opinion only.
but they never think true or false ...
i respect all of them...
I believes whole universe depending
on true ...থিউরি
Govinda
hello
Jackson
hi
Elyana
we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe
there many theory to born universe but what is the reality of big bang theory to born universe
Asmit
what is the exact value of π?
Nagalakshmi
by big bang
universal
there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.
Aarya
I think after Big Bang!
Michele
from where on earth could u observe all the stars during the during the course of an year
is that so. the question was in the end of this chapter
Karuna
in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*).
So it would just barely creep over the horizon.
Christopher
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