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In addition to the tutorial for Vector Product of Two Vectors on this page, you can also access the following Vectors and Scalars learning resources for Vector Product of Two Vectors
|2.5||Vector Product of Two Vectors|
In our previous Physics Tutorial, we discussed the "Dot (scalar) product" of two vectors. It was explained that when two vectors are multiplied in scalar mode, the result is a scalar (number). This seems a bit strange to understand, since we expect the product to be a vector considering the "closure" property of multiplication (i.e. the product of a multiplication belongs to the same set or category as the two factors). However, we saw that this is possible in dot multiplication of vectors.
But, now that we have explained the meaning of dot product, it is much simpler to understand the other type of vectors multiplication, i.e. the cross (vector) product of two vectors.
Symbolically, the cross product of two vectors a⃗ and b⃗ is denoted by the symbol (×). Geometrically, it represents a new vector, which is perpendicular to the plane on which the two vectors lie.
Mathematically, the cross product magnitude of two vectors represents the magnitude of the surface area enclosed by the two vectors a⃗ and b⃗ and their parallel extensions (the area of the parallelogram formed by the two vectors a⃗ and b⃗).
From geometry, it is known that the area of a parallelogram is calculated by the formula
Look at the figure below:
From Trigonometry, it is known that
Here, Opposite side of triangle = Height of parallelogram and Hypotenuse = Lateral side of parallelogram. Therefore, we obtain
Substituting the Base and Lateral side with the lengths of vectors a⃗ and b⃗ respectively as shown in the figure below,
But we stated before that
and since it is obvious that the above equation is true for magnitudes as well, i.e.
It seems as a kind of shortcoming the fact that by formula we can only calculate the magnitude of the vector product obtained by the cross product of two vectors. However, this issue is fixed by applying the "drill rule" (or the screwdriver rule). You may remember that if you have ever turned the screwdriver in the clockwise direction, the screw has moved linearly away from you and when you have tried to remove the screw, you have turned the screwdriver anticlockwise to make the screw move towards you. This is illustrated in the figure below:
The same idea is used to determine the direction of the vector c⃗ which is the cross product vector of a⃗ and b⃗. Thus, if we have to find the direction of c⃗ = a⃗ × b⃗, we start rotating from a⃗ to b⃗ (in our example this direction is anticlockwise). Therefore, (like in the screw) the direction of the vector c⃗ will be upwards as shown in the figure.
On the other hand, if we have to find the cross product d⃗ = b⃗ × a⃗ we must start rotating from b⃗ to a⃗. In our example, such direction is clockwise and based on the screwdriver rule, the direction of the vector d⃗ will be downwards. Look at the figure:
From the two above figures, it is obvious that the vectors c⃗ and d⃗ are opposite. Hence, we can write
If the coordinates of the vectors a⃗ and b⃗ (namely xa, ya, za, xb, yb and zb) are given, we can find the coordinates of the vector c⃗ = a⃗ × b⃗ (i.e. xc, yc and zc) using the following formulae:
Three forces are acting on the same object placed at the origin of the coordinates system. The tip of the first force is at (-2, 3, 0) and that of the second force is at (1, 5, -4). What are the coordinates of the tip of the third force vector if these three forces obey the rule of vectors' cross production?
Before starting the calculation, the position of the two vectors F⃗1 and F⃗2 is as shown below
Thus, the direction of the third fore F⃗3 will be determined by the cross product of the tip's coordinates of the two forces F⃗1 and F⃗2. We have
Substituting the known values (F1x = -2, F1y = 3, F1z = 0, F2x = 1, F2y = 5 and F2z = -4), we obtain for the tip's coordinates of F3
Therefore, the tip of vector F3 will be at (-12, -8, -13). This is illustrated in the figure below.
In Physics, there are a lot of applications of vector cross product. They are much more than dot product applications. Let's discuss briefly some of them.
1. Moment of force M⃗ as cross product of Force F⃗ and linear distance from the turning point Δx⃗
It is obvious Moment of force is a vector quantity as unlike Work, it is obtained through the cross product of Force and Linear distance from the turning point.
2. Magnetic force F⃗ of a conductor at rest as a cross product of Magnetic induction B⃗ and the conductor length L⃗ multiplied with the scalar I (current).
3. Magnetic force F⃗ of a moving conductor as a cross product of Magnetic induction B⃗ and Velocity v⃗ multiplied with the scalar q (electric charge).
4. The angle between two forces F⃗1 and F⃗2 can be calculated using the cross product if the magnitudes of the two vectors F⃗1 and F⃗2 and that of F⃗1 × F⃗2 are known,
and so on.
A conducting wire is placed between the two poles of a horseshoe magnet as shown in the figure. The magnetic field lines (the induction B⃗) lie from the North to the South pole of the magnet. Electric charges flow through the conducting wire in a direction that is away from us (from us to the sheet). To make the reader have a better idea, the figure is slightly inclined.
If the magnitude of the magnetic induction is b⃗ = 4 Tesla (B⃗ = 4 T) and the amount of electric charges flowing through the wire is q = 6 Coulombs (q = 6 C), find the velocity v⃗ of wire (both magnitude and direction) if it forms a right angle to the magnetic field lines. The magnetic force produced is F⃗ = 0.2N.
From the figure, it is easy to see that we move clockwise from q-direction to B-direction. Therefore, if you consider the wire as the handle of a screwdriver, and giving that you are rotating it clockwise, the screw will move forward. Thus, in this case, the wire will move due left.
Also, from the clues, it is obvious that the wire is perpendicular to the magnetic field lines. This means sin θ = sin 900 = 1.
The magnitude of velocity is calculated by the equation
From the cross product rules, we have
Substituting the known values, we obtain (giving that sin 90° = 1)
As we discussed in the previous example, we may have to multiply the cross product of two vectors with a scalar. The result is still a vector as the cross product gives a vector and the product of a vector by a scalar gives a vector as well.
Another rule for finding the direction of the cross product vector
In the cross product
we can apply the "right hand rule" to find the direction of the vector product c⃗ when the directions of a⃗ and b⃗ are known. Thus, if we put the index and middle fingers of the right hand as shown in the figure,
we obtain the direction of vector
based on the drill (screwdriver) rule. The thumb shows the direction of the product vector.
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