Vector Direction
A vector is a quantity, which has both magnitude and direction. Examples of
vectors include displacement, velocity, acceleration, and force. To fully describe
one of these vector quantities, it is necessary to tell both the magnitude and the
direction. For instance, if the velocity of an object were said to be 25 m/s, then
the description of the object's velocity is incomplete; the object could be moving
25 m/s south, or 25 m/s north, or 25 m/s
southeast. To fully describe the object's
velocity, both magnitude (25 m/s) and direction
(e.g., south) must be stated.
In order for such descriptions of vector
quantities to be useful, it is important that
everyone agree upon how the direction of an
object is described. Most of us are accustomed
to the idea that "up" on a map refers to the
"northward" direction and "right" on a map
refers to the "eastward" direction. This is a
mere convention which map-makers have used
for years and upon which we all can agree. But
what is the direction of a vector quantity is not due north or due east but
somewhere in between north and east? For such cases as this, it is important that
there be some convention for describing the direction of such a vector. We use
the bearing method that has North at 0, with the directions rotating clock –
wise.
See http://virtualskies.arc.nasa.gov/main/mnavigation.html for more info on this and airplane navigation.
See the animation VecDirect
Vector Addition: 6 + 8 = ?
Most of us are accustomed to the following form of mathematics:
6 + 8 = 14.
Yet, we are extremely uneasy about this form of mathematics:
6 + 8 = 10 and
6+8=2
and
6 + 8 = 5.
When we become students of physics and approach the task of adding vector
quantities, we soon become aware of the fact that the addition of two vector
quantities with magnitudes of 6 and 8 will not necessarily result in an answer of
14. The rules for adding vector quantities are different from the rules for adding
two quantities arithmetically. Thus, 6 + 8 will not necessarily be equal to 14.
Vectors are quantities that include a direction. As such, the addition of two or
more vectors must take into account that the quantities being added have a
directional characteristic. There are a number of methods for carrying out the
addition of two (or more) vectors. The most common method is the "head-to-tail"
method of vector addition. Using such a method, the first vector is drawn to scale
in the appropriate direction. The second vector is then drawn such that its "tail" is
positioned at the "head" (vector arrow) of the first vector. The sum of two such
vectors is then represented by a third vector, which stretches from the tail of the
first vector to the head of the second vector. This third vector is known as the
"resultant" - it is the result of adding the two vectors. The resultant is the vector
sum of the two individual vectors. Of course, the actual magnitude and direction
of the resultant is dependent upon the direction which the two individual vectors
have.
This principle of the "head-to-tail" addition of vectors is illustrated in the
animation below. In each frame of the animation, a vector with magnitude of 6 (in
green) is added to a vector with magnitude of 8 (in blue). The resultant is
depicted by a black vector, which stretches from the tail of the first vector (8
units) to the head of the second vector (6 units).
VECCADD
As can be seen from this animation, 8 + 6 could be equal to 14, but only if the
two vectors are directed in the same direction. All that can be said for certain is
that 8 + 6 can add up to a vector with a maximum magnitude of 14 and a minimum
magnitude of 2.
Vector Addition: The Order Does NOT Matter
The summative result of two or more vectors can be determined by a process of
vector addition.
The most common method of adding vectors is the graphical method of
head-to-tail addition. This method involves the selection of a scale (e.g., 1 cm =
5 km), and the subsequent drawing of each vector to scale in the specific
direction. The tail of each consecutive vector begins at the head of the most
recent vector. The resultant vector (the summative result of the addition of the
given vectors) is then drawn from the tail of the first vector to the head of the
last vector. The magnitude and direction of the resultant is then determined using
a protractor, ruler, and the indicated scale.
The question often arises to the importance of the order in which the vectors are
added. For instance, if five vectors are added - let's call them vectors A, B, C, D
and E - then will a different resultant be obtained if a different order of addition
is used. Will A + B + C + D + E yield the same result as C + B + A + D + E or
D + E + A + B + C? The animation below provides the answer.
Observe the animation a couple of times and see what the answer is.
(AO.GIF)
As shown in the animation above, the order in which two or more vectors are
added does not affect the outcome. Adding A + B + C + D + E yields the same
result as adding C + B + A + D + E or D + E + A + B + C! The resultant, shown
as the green vector, has the same magnitude and direction regardless of the order
in which the five individual vectors are added.
Draw the picture here.
The Plane and The Wind
Observe the three planes in the animation below. (PL1.GIF) Each plane is heading
south with a speed of 100 mi/hr. Each plane flies amidst a wind, which blows at
20 mi/hr. In the first case, the plane encounters a tailwind (from behind) of 20
mi/hr. The combined effect of the tailwind and the plane speed provide a resultant
velocity of 120 mi/hr.
In the second case, the plane encounters a headwind (from the front) of 20 mi/hr.
The combined effect of the headwind and the plane speed provide a resultant
velocity of 80 mi/hr. In the third case, the plane encounters a crosswind (from the
side) of 20 mi/hr. The combined effect of the headwind and the plane speed
provide a resultant velocity of 102 mi/hr (directed at an 11.3 degree angle east of
south). These three resultant velocities can be determined using simple rules of
vector addition. In the case of the crosswind, the Pythagorean Theorem and SOH
CAH TOA are utilized to determine the magnitude and the direction of the
resultant velocity.
(PL.GIF)
Draw it.
The River Boat
Observe the motion of the riverboats below. In each case, the boat is heading
across the river. In the top case, the presence of a current causes the boat to
travel downstream as it simultaneously moves across the river; the result is that
it reaches the opposite shore at a point downstream from where it started. In the
bottom case, the absence of a current means that the boat moves straight across
the river, it ultimately reaches a point on the opposite shore directly across from
where it started. It is important to note that the current is what carries the boat
down the stream; it is not the motor of the boat that provides its downstream
motion.
(RB.GIF)
Assuming that in each case the motor of the boat propels it across the river with
the same force, in which case (with or without a current) will the boat make it
across the shore the soonest? Make your observation in the above animation to
find out. You might observe something other than you expect.
Draw it
For more info see web site
http://www.glenbrook.k12.il.us/gbssci/phys/Class/BBoard.html
Select Vectors & Projectiles