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In this section you will:
  • View vectors geometrically.
  • Find magnitude and direction.
  • Perform vector addition and scalar multiplication.
  • Find the component form of a vector.
  • Find the unit vector in the direction of  v .
  • Perform operations with vectors in terms of  i  and  j .
  • Find the dot product of two vectors.

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, as shown in [link] . What are the ground speed and actual bearing of the plane?

Image of a plan flying SE at 140 degrees and the north wind blowing

Ground speed refers to the speed of a plane relative to the ground. Airspeed refers to the speed a plane can travel relative to its surrounding air mass. These two quantities are not the same because of the effect of wind. In an earlier section, we used triangles to solve a similar problem involving the movement of boats. Later in this section, we will find the airplane’s groundspeed and bearing, while investigating another approach to problems of this type. First, however, let’s examine the basics of vectors.

A geometric view of vectors

A vector    is a specific quantity drawn as a line segment with an arrowhead at one end. It has an initial point    , where it begins, and a terminal point    , where it ends. A vector is defined by its magnitude    , or the length of the line, and its direction, indicated by an arrowhead at the terminal point. Thus, a vector is a directed line segment. There are various symbols that distinguish vectors from other quantities:

  • Lower case, boldfaced type, with or without an arrow on top such as v , u , w , v , u , w .
  • Given initial point P and terminal point Q , a vector can be represented as P Q . The arrowhead on top is what indicates that it is not just a line, but a directed line segment.
  • Given an initial point of ( 0 , 0 ) and terminal point ( a , b ) , a vector may be represented as a , b .

This last symbol a , b has special significance. It is called the standard position    . The position vector has an initial point ( 0 , 0 ) and a terminal point a , b . To change any vector into the position vector, we think about the change in the x -coordinates and the change in the y -coordinates. Thus, if the initial point of a vector C D is C ( x 1 , y 1 ) and the terminal point is D ( x 2 , y 2 ) , then the position vector is found by calculating

A B = x 2 x 1 , y 2 y 1 = a , b

In [link] , we see the original vector C D and the position vector A B .

Plot of the original vector CD in blue and the position vector AB in orange extending from the origin.

Properties of vectors

A vector is a directed line segment with an initial point and a terminal point. Vectors are identified by magnitude, or the length of the line, and direction, represented by the arrowhead pointing toward the terminal point. The position vector has an initial point at ( 0 , 0 ) and is identified by its terminal point a , b .

Find the position vector

Consider the vector whose initial point is P ( 2 , 3 ) and terminal point is Q ( 6 , 4 ) . Find the position vector.

The position vector is found by subtracting one x -coordinate from the other x -coordinate, and one y -coordinate from the other y -coordinate. Thus

v = 6 2 , 4 3 = 4 , 1

The position vector begins at ( 0 , 0 ) and terminates at ( 4 , 1 ) . The graphs of both vectors are shown in [link] .

Plot of the original vector in blue and the position vector in orange extending from the origin.

We see that the position vector is 4 , 1 .

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
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"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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