v1A.gif (1045 bytes)

Vectors

v2A.gif (1070 bytes)
h2A.gif (678 bytes)h2A.gif (678 bytes)

Note: Vectors are indicated using the "bold-letter" notation.


Previously Asked Questions

Q:    How do you geometrically subtract two vectors?

A:    If you have two vectors a and b and want to subtract them.  First you need to reverse the direction of vector b so that it becomes vector -b.  Then you would take vector -b and  place its tail at the end of vector a.  Then you would draw a line from the tail of a to the head of -b.   This line is your new vector with its head at the head of -b.

Example: If we have vector a and want to subtract vector b.  First we reverse the direction of b.   This will give us -b which we can then add to a.  

   b.gif (267 bytes) =>  b-.gif (284 bytes)

Now we place the tail of vector -b at the head of vector a.   A new vector, d, can be drawn from the tail of vector a to the head of vector -b.

a.gif (315 bytes)  +  b-.gif (284 bytes) =   d.gif (684 bytes)

 

Q:    How do you geometrically add two vectors?

A:    If you have two vectors a and b and want to add them.  Then you would take vector b and place its tail at the end of vector a.  Then you would draw a line from the tail of a to the head of b.   This line is your new vector with its head at the head of b.

Example: If we have a vector a and want to add it to a vector b.  We place the tail of vector b at the head of vector a.  A new vector, c, can be drawn from the tail of vector a to the head of vector b.

a.gif (315 bytes)  +  b.gif (267 bytes) =   c.gif (918 bytes)

 

Q:     What is the difference between a scalar and a vector?

A:     Scalar have magnitude only.  Vectors have a magnitude plus a direction.

Example: Temperature and speed are scalars because they only have a magnitude.  Velocity and displacement are vectors because they have a magnitude and a direction.

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References

Components of a Vector

components.gif (2726 bytes)

     A two dimensional vector a has two components, ax and ay, the values for these components are found by drawing lines from the end of a to the coordinate axis.  Thus
    ax  = a cos theta2.gif (833 bytes)    and   ay = a sin theta2.gif (833 bytes)
where theta2.gif (833 bytes) is the angle from the positive direction of the x-axis to the direction of a

Given the components of a vector we can reconstruct the vector from

3-6.gif (1262 bytes)

 

Unit Vectors

unitA.gif (4783 bytes)

     Unit vectors are vectors with a magnitude of one and pointing in a particular direction.  The sole purpose of unit vectors is specify direction.  In the Three Dimensional Cartesian System:

        ihat.gif (56 bytes)  is the unit vector in the positive x direction
        jhat.gif (61 bytes)  is the unit vector in the positive x direction
        khat.gif (62 bytes)  is the unit vector in the positive x direction

       A vector can be decomposed into its ihat.gif (56 bytes), jhat.gif (61 bytes), and khat.gif (62 bytes) components by breaking the vector into magnitudes on the x, y, and z axis respectfully.

 

[Top] [Previously Asked Questions] [References]


Mechanics List of Topics

Measurements Newton's Laws Potential Energy and Conservation of Energy Rotation of
Rigid Bodies
Elasticity
Vectors Forces and Fields Linear Momentum Angular Momentum Mechanical
Oscillations
Motion of Point-Mass Objects in One Dimension The Gravitational Field Collisions Torque Mechanical Waves
Motion of Point-Mass Objects in Two and Three Dimensions Kinetic Energy
and Work
Circular Motion of Point-Mass Objects Equilibrium Sound

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