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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

given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
Snalo Reply
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Mark Reply
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Miranda
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Miranda Drice
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jai
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Miranda
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jai
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Miranda
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Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
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Propessor
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Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
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Propessor
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What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
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Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
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Miranda
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Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
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Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
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Miranda
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Nelson Reply
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Miranda
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Miranda
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Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
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cos(- z)=cos z
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what is a algebra
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(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
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Nokwanda
C'est comment
Abdel
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
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Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
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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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