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We can check our earlier projection value from a different viewpoint now that we know the angle between the vectors.
From the drawing on your graph board, you should see that vector A forms the hypotenuse of a right triangle and theprojection of vector A onto vector B forms the base of that triangle. You should know from the earlier module on trigonometry that the length of the base is
base = Amag * cos(angle), or
base = 2 * cos(44.3 degrees), or
base = 1.43 units
which matches the length of the projection that we computed earlier.
Figure 1 contains some facts worth remembering about the vector dot product.
| Figure 1 . Facts worth remembering for the dot product. |
|---|
Given two vectors, A and B with their tails at the origin and
their tips at ax, ay, bx, and by respectively(A dot B) = (ax * bx) + (ay * by)
where(A dot B) represents the dot product of the vectors
named A and Balso
(A dot B) = Amag*Bmag*cosine(angle between A and B)where
Amag is the magnitude of vector ABmag is the magnitude of vector B
The projection of A onto B = (A dot B)/(Bmag)The angle between A and B = arccos((A dot B)/(Amag *Bmag))
For a given pair of vectors, the dot product can be thoughtof as a measure of the extent to which they are parallel.
The closer they are to parallel, the greater will be thevalue of the dot product. |
Let's begin our discussion of the vector cross product with some background information.
The cross product , sometimes called a vector product , is an operation on two vectors in three-dimensional space. The operation results in a vector that is perpendicular to both of the vectors being multiplied.
The name of the operation
The name "cross product" derives from the fact that a special character that looks likean "x" is often used to indicate the nature of the operation.
I doubt that the special character will display properly on your Braille display. In this module, therefore, I will use an actual "x" character instead of the special character that istypically used. For example, I will indicate the cross product between vectors A and B as
AxB
A cross product with a zero result
If either of the vectors being multiplied has a magnitude of 0, the cross product will be zero. Also if the vectors being multiplied are parallel, theircross product will be zero.
The area of a parallelogram
Except for the case of perpendicular vectors, the magnitude of the cross product between two vectors equals the area of a parallelogram with the vectorsforming two sides of the parallelogram. For the case of perpendicular vectors, the parallelogram becomes a rectangle andthe magnitude of the product is the area of that rectangle.
The direction of the resultant vector
As mentioned earlier, the result of the cross product is a vector that is perpendicular to both of the vectors being multiplied. The resultant vector cansatisfy that requirement and point in ether of two directions. The actual direction depends on certain orientation conventions as described by the right-hand rule .
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