Component Form and Magnitude

A vector is a quantity with both magnitude and direction. To differentiate vectors from variables, constants and imaginary numbers vectors are denoted by a lowercase boldface variable, v, or a variable with a harpoon arrow above it, v .

In this discussion vectors will be denoted by lowercase boldface variables. Examples of vector quantities are velocity, acceleration and force.

Figure 1 shows two vectors v and u as directed line segments.



The vector v is represented by the directed line segment RS and has an initial point at R and a terminal point at S.

Vectors are often represented in component form. Vector v in Figure 1 has an initial point at the origin (0,0) and is said to be in standard position. A vector in standard position can be represented by the coordinates of its terminal point. Thus vin component form = v 1 , v 2 .

Notice that the brackets surrounding the vector components v1 and v2 are pointed not round like parentheses.


The magnitude of v or RS is represented by RS or v and is calculated using the Distance Formula, v= v 1 2 + v 2 2 .

MAGNITUDE OF A VECTOR:

v= v 1 2 + v 2 2



Vector u represented by GH in Figure 1 is not in standard position as its initial point is not the origin (0,0). The component form and magnitude of vector u can be calculated as follows:

COMPONENT FORM OF DIRECTED LINE SEGMENT:


Initial point:         G (g1, g2)
Terminal Point:     H (h1, h2)


GH = h 1 g 1 , h 2 g 2  = u 1 , u 2 =u



MAGNITUDE OF DIRECTED LINE SEGMENT:


Initial point:         G (g1, g2)
Terminal Point:     H (h1, h2)


GH = ( h 1 g 1 ) 2 + ( h 2 g 2 ) 2 = u 1 2 + u 2 2



Let's look at a couple examples.

Example 1:      Find the component form and magnitude of vector u in Figure 1.

Step 1: Identify the initial and terminal coordinates of the vector.

Initial Point G: (-2, 2)


Terminal Point H: (-4, 4)

Step 2: Calculate the components of the vector.


Subtract the x-component of the terminal point from the x-component of the initial point for your x-component of the vector. Do the same for the y-components.

u= h 1 g 1 , h 2 g 2  = u 1 , u 2


u=4( 2 ), 42


2, 2=u

Step 3: Calculate the magnitude of the vector.

||u|| = u 1 2 + u 2 2


||u|| = ( 2 ) 2 + ( 2 ) 2


4+4


8

Example 2:      Find the component form and magnitude of vector v in standard position with terminal point (8, -2).

Step 1: Identify the initial and terminal coordinates of the vector.


Because vector v is in standard position it's initial point is (0,0)

Initial Point: (0, 0)


Terminal Point: (8, -2)

Step 2: Calculate the components of the vector.


Subtract the x-component of the terminal point from the x-component of the initial point for your x-component of the vector. Do the same for the y-components.


In this case the vector is in standard form therefore the components of the vector are the same as the components of the terminal point.

v= v 1 0, v 2 0 = v 1 , v 2


v=80, 20


8,2=v

Step 3: Calculate the magnitude of the vector.

||v|| = v 1 2 + v 2 2


||v|| = ( 8 ) 2 + ( 2 ) 2


64+4


68


2 17





Related Links:
Math
algebra
Scalar Multiplication and Vector Addition
Unit Vectors
Pre Calculus


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