Component Vectors

Key Questions

  • How do you find the resultant as a sum of two components?

    Answer:

    " "
    Please read the explanation.

    Explanation:

    " "
    How do we use the components of two vectors to find the resultant vector by adding the two vectors ?

    A Vector is defined as a quantity with both magnitude and direction.

    Two vectors are shown below:

    color(red)(vec(OA) and vec(OB)

    We will also be using these vectors in our example later.

    enter image source here

    vec(OA) = hat(u)=(2 hat i+5 hat j)

    In component form

    hat(u)=<2,5>

    vec(OB) = hat(v)=(4 hat i-8 hat j)

    In component form

    hat(v)=<4,-8>

    Let us see how we can add these two vectors:

    hat (u) + hat (v) = (2 hat i+5 hat j)+(4 hat i-8 hat j)

    Using component form:

    hat (u) + hat (v) = <2 ,5 >+<4-8 >

    Add color(red)(i components and color(red)(j components together:

    hat (u) + hat (v) = <2+4>+<5-8 >

    color(red)(hat (u) + hat (v) =<6, -3>

    We can represent this solution graphically as follows:

    enter image source here

    The solution is represented by

    color(red)(w=hat (u) + hat (v) =<6, -3>

    OR

    color(red)(w=hat (u) + hat (v) =(6i -3j)

    Note: Alternative graphical solution process:

    vec(OA) can also be translated to the line in green (BC).

    OR

    vec(OB) can be translated to the line in blue (AC).

    We can see that color(red)(w is the solution.

    Hope it helps.

    Sridhar V. · 1 · Aug 4 2018
  • How do you use vector components to find the magnitude?

    To find the magnitude of a vector using its components you use Pitagora´s Theorem.

    Consider in 2 dimensions a vector vecv given as:
    vecv = 5veci + 3vecj (where veci and vecj are the unit vectors on the x and y axes)
    enter image source here
    The magnitude of this vector (or its length in geometrical sense) is given using Pitagora's Theorem, as:
    |vecv| =sqrt(5^2+3^2)= 5,8

    The same thing applies in 3 dimensions, the only thing is to include the third component.

    So if the vector is now given as:
    vecv = 5veci+ 3vecj + 2veck
    The magnitude will be:
    |vecv|= sqrt(5^2+3^2+2^2) = 6,2

    Gió · 1 · Dec 14 2014
  • How do you use component vectors to solve real-world and applied problems?

    Often when two processes interact we only know the component vector values and need to be able to combine these to get a desired result.

    This might be more easily understood by an example:
    Suppose I am trying to fly from point A to point B which is due North of point A. My plane flies at an air speed of 100 miles/hour but there is a wind blowing due West at 30 miles/hour. How many degrees East of North do I need to orient my plane to fly in a straight line to B?

    enter image source here

    From the above diagram, I need to head my plane (approximately)
    17.5^o East of North.

    This problem could be extended to ask:
    If it is 200 miles from A to B and my plane has enough gas to fly 250 miles will I be able to make this trip?

    Alan P. · 1 · Feb 11 2015
  • What are component vectors?

    A vector has both magnitude (which is its length) and direction (which is its angle).

    Any two dimentional vector at an angle will have a horizontal and a vertical component .
    A vector written as ( 12 , 8 ) will have 12 as its horizontal component, and 8 as its vertical component, and because both components are positive, the vector is pointing to the northeastern direction.

    physicsclassroom

    Samantha C. · 1 · Jan 10 2015

Questions

Triangles and Vectors