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You can see that the value of the dot product of the two vectors in Figure 10 is 0.0, and the angle between the two vectors is 90 degrees. Therefore,although the magenta vector in Figure 10 is much longer than the magenta vector in Figure 8 , the magenta vector in Figure 10 is still perpendicular to the black vector. Thus, Figure 8 and Figure 10 show two of the infinite number of magenta vectors that are perpendicular to the black vector in those images.
The dot product of perpendicular vectors in 3D
As I mentioned earlier, the topic of perpendicularity in 3D is more complicated than is the case in 2D. As is the case in 2D, there are an infinite numberof vectors that are perpendicular to a given vector in 3D. In 2D, the infinite set of perpendicular vectors must have different lengths taken inpairs, and the vectors in each pair must point in opposite directions.
An infinite number of perpendicular vectors having the same length
However, in 3D there are an infinite number of vectors having the same length that are perpendicular to a given vector. All of the perpendicular vectorshaving the same length must point in different directions and they must all lie in a plane that is perpendicular to the given vector.
Perpendicular vectors having different lengths may point in the same or in differentdirections but they also must lie in a plane that is perpendicular to the given vector.
A wagon-wheel analogy
Kjell explains the situation of an infinite set of 3D vectors that are perpendicular to a given vector by describing an old-fashioned wagon wheel withspokes that emanate directly from the hub and extend to the rim of the wheel. The hub surrounds the axle and each of the spokes is perpendicular to the axle.Depending on the thickness of the spokes, a large (but probably not infinite) number of spokes can be used in the construction of the wagon wheel.
Another wheel at the other end of the axle
In this case, the wagon wheel lies in a plane that is perpendicular to the axle. There is normally another wheel at the other end of the axle.Assuming that the axle is straight, the second wheel is in a different plane but that plane is also perpendicular to the axle. Thus, the spokes in thesecond wheel are also perpendicular to the axle.
If there were two identical wheels at each end of the axle for a total of four wheels (the predecessor to the modern 18-wheel tractor trailer) , the spokes in all four of the wheels would be perpendicular to the axle. Again, the point isthat there are an infinite number of vectors that are perpendicular to a given vector in 3D.
A general formulation of 3D vector perpendicularity
By performing some algebraic manipulations on the earlier 3D formulation of the dot product, we can develop the equations shown in Figure 11 that describe an infinite set of vectors that are perpendicular to a givenvector.
| Figure 11 . A general formulation of 3D vector perpendicularity. |
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dot product = x1*x2 + y1*y2 + z1*z2
If the two vectors are perpendicular:x1*x2 + y1*y2 + z1*z2 = 0.0
x1*x2 = -(y1*y2 + z1*z2)x2 = -(y1*y2 + z1*z2)/x1
ory2 = -(x1*x2 + z1*z2)/y1
orz2 = -(x1*x2 + y1*y2)/z1 |
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