AERONAUTICS - Flight is a magnificent human achievement, and some flight requires no engines at all.  An advanced glider, known as a sailplane, has high maneuverability and glide capabilities.  The Ventus 2B sailplane placed first at the World Soaring Contest (yes, there is one) in New Zealand.  

At one competition, a sailplane traveled forward at a rate of 8 meters per second (mps), and it descended at a rate of 4 mps.  Later, you will use vector methodology to determine the 

  A vector can be represented geometrically as a 

 line segment.  The length of this line segment represents the 

 of the vector, and the arrow on the segment describes the 

.  Segments can be labeled classically, given the symbol “

” to further describe it as part of a vector.  The magnitude of a segment ⇀a is always a positive value, so the length can be considered |⇀a| .

As you can see, much can be gleaned from the physical representation of the vectors in space.  When comparing magnitude and direction together, you can make basic claims about two different vectors.  When given, two vectors are equal 

 they have the same magnitude and direction.

Picture a vector with its initial point at the origin of a coordinate plane.  This vector is considered to be in 

 position, and its direction is described as the angle formed between the positive x-axis and the vector being the terminal side.

In this diagram, you can use Trigonometry to find the measure of the direction of the vector.  With the adjacent side of 3 and opposite side of 2, the vector direction would be described as the tan

State whether or not the two given vectors are equal, and compare the magnitude and direction.

The sum of two or more vectors is called the 

of the vectors.  There are two methods with which one can use to find the value of the resultant.  The first is the Parallelogram Method.  This geometric shape is used because it allows for a combination of two vectors regarding magnitude and direction.

In this image, ⇀P and ⇀Q have different magnitudes and directions, and are therefore not equal to one another.  Because of this, they can be combined to find a resultant.  When placed together at the same initial point, a shape can be drawn by measuring the magnitude of each vector and drawing a parallel segment opposite the given vector.  Segment BC is the “imaginary” side drawn opposite ⇀P, for instance.  Once drawn, a diagonal can be drawn from the initial point to where the two “imaginary” sides meet.  If you measure that diagonal, that is the measurement of the resultant’s magnitude.

f) Looking back on the sailplane example, find the magnitude of the resultant velocity in meters per second.  Draw a diagram to represent the vectors.  Let 1 cm = 2 mps.

Another way to calculate the resultant of two vectors is the 

In this example, ⇀P and ⇀Q are not equal, as they have differences in direction.  To find the resultant, you can copy ⇀Q so that its initial point lies on the terminal point of ⇀P.  Connect the initial point of ⇀P to the terminal point of ⇀Q, here creating segment AC, or ⇀P + ⇀Q.  

g) Using the same information from the last prompt involving the sailplane, use the triangle method to calculate the resultant of the vectors.  Use the same scale.  Compare your results to f).

Two or more vectors whose sum is a given vector are called 

 of the given vector.  Components can have any direction.  Often, it can be useful to use two perpendicular components in order to find a resultant, or to break a resultant down.  When one component is moving away from the other, they are both considered positive.  When one component is moving in a different direction, or towards the other component, it is considered negative, and the larger positive.

You can use this relationship to create a right triangle, which can be used in any way that a right triangle can be used.  One useful method is Trigonometry!

You can see in this additional example that you can use a given magnitude of a vector and these relationships to gain more information about the components.  The sine of a created or given angle or direction will give the opposite side, and the cosine the adjacent.  This also works with one missing component and the missing vector magnitude, but given a directional angle.

h) A ship leaving port sails for 75 miles in a direction 35° north of due east.  Find the magnitude of the vertical and horizontal components in miles.

Finally, you can also manipulate the magnitude of a single vector while using a 

, which acts much as it does in coordinate Geometry.  This use of a scalar is used in comparing vectors, their sums, and their magnitudes.

a diagram using the triangle method to find the magnitude and direction of 2(⇀v) - ½(⇀w).  The original measure of ⇀v is 3 cm and ⇀w is 8 cm.  

j) A piling (yes, a piling!) for a high-rise building is pushed by two bulldozers at exactly the same time.  One bulldozer exerts a force of 1550 pounds in a westerly direction.  The other bulldozer pushes the piling with a force of 3050 pounds in a northerly direction.  If a resultant (r) represents the total force upon the piling, it must be the |r|.  What is this magnitude?  What is the direction?  Round answers to the nearest 10 pounds of force.

🔑Use the following magnitude and direction for the vectors to solve: ⇀r = 4 cm, 135°  , ⇀s =2.3cm, 20°,    
                              ⇀t = 5 cm, 85°,  ⇀u = 3.1 cm, 94°

p) Police Officer Patricia Malloy is investigating an automobile accident.  A car slid off an icy road at a 40° angle from the center line of the road with an initial velocity of 47 mph.  What were the initial horizontal and vertical components of the car’s velocity?

q) Matsuko has attached two wires to a corner fence post to complete a horse paddock.  The wires make an angle of 90° with each other.  If each wire pulls on the post with a force of 50 pounds, what is the resultant force acting on the pole?

r) Belkis is pulling a toy by exerting a force of 1.5 N on a string attached to the toy.  The string makes an angle of 52° with the floor.  What are the vertical and horizontal components of this force?

r.5) If little Belkis raises the string so that it makes a 78° angle with the floor, what are the magnitudes of the horizontal and vertical components of the force?  Compare the results to find the more efficient floor-based angle of pull.

s) 🦄 Mrs. Keaton is the director of a local art gallery.  She needs to hang a painting weighing 24 pounds with a wire whose parts form a 120° angle.  Find the pull of each part of the wire so Mrs. Keaton can find the correct gauge.

Students will use Geometric concepts and Trigonometry to  calculate movement and magnitude in space.

Vectors in Space