| << Chapter < Page | Chapter >> Page > |
There is more than one way to multiply vectors (see (External Link) ). I will explain two of those ways in this module:
I will begin with some background information on the dot product of two vectors.
The terms dot product, inner product, and scalar product all mean the same thing and are used in various context's by different authors.
The term dot product derives from the fact that a vector product of this sort is often written as the names of the two vectors separated by a dot.However, that special dot character is probably not compatible with your Braille display. Therefore, I will write the dot product of the vectors named A and B as
(A dot B)
The term scalar product derives from the fact that a vector product of this sort more closely resembles scalar arithmetic than vector arithmetic. Inparticular, unlike the cross product (that will be discussed later), the result of the dot product does not have a direction.
In order for you to better understand the nature of a vector dot product, I recommend that you create a Cartesian coordinate system on your graph board, and draw thefollowing two vectors.
A vector diagram for your graph board
Draw the first vector from the origin to a point at
x = 1y = 1.73
Label this vector A.Draw a second vector from the origin to a point atx = 2.9
y = 0.78Label this vector B. References to the vector coordinates
I will refer to the coordinates at the tip of vector A as ax and ay. Similarly, I will refer to the coordinates at the tip of vector B as bx and by.
Using this nomenclature , the dot product of any two vectors is given by
(A dot B) = (ax * bx) + (ay * by)
where
So what?
By now you are probably saying "So what? Why should I care?"
Although it isn't obvious from what you see above, the dot product of two vectors is also equal to the product of their magnitudes andthe cosine of the angle between them. In other words,
(A dot B) = Amag*Bmag*cosine(angle between A and B)
where
If you divide the dot product of A and B by the magnitude of B, the result is equal to the projection of vector A ontovector B. In other words,
(A dot B)/(Bmag) = projection of A onto B
This sort of projection operation is an operation that occurs frequently in physics. For example, the horizontal component of a velocity vector is the projection ofthe velocity vector onto the horizontal axis. Similarly, the vertical component of a velocity vector is the projection of the velocity vector onto the verticalaxis.
Let's work through some numbers
Substituting your coordinate values into the expression given above yields
(A dot B) = (ax * bx) + (ay * by), or
(A dot B) = (1.0*2.9) + (1.73*0.78), or
Notification Switch
Would you like to follow the 'Accessible physics concepts for blind students' conversation and receive update notifications?
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|