Cross (Vector) Product of Two Vectors

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2.5Vector Product of Two Vectors

p>In this Physics tutorial, you will learn:

  • The meaning of cross product of two vectors
  • What does the cross product represent geometrically?
  • How to calculate the cross product of two vectors?
  • How to find the direction of cross product vector?
  • How to calculate the cross product in coordinates?
  • Which are some applications of cross product in Physics?

Introduction

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.

The meaning of cross product of vectors multiplication

Symbolically, the cross product of two vectors a and b is denoted by the symbol (×). Geometrically, it represents a new vector, which is perpendicular to the plane on which the two vectors lie.

Physics Tutorials: This image shows

Mathematically, the cross product magnitude of two vectors represents the magnitude of the surface area enclosed by the two vectors a and b and their parallel extensions (the area of the parallelogram formed by the two vectors a and b).

Physics Tutorials: Vector Product of Two Vectors

How to calculate the cross product of two vectors?

From geometry, it is known that the area of a parallelogram is calculated by the formula

Area = Base × Height

Look at the figure below:

Physics Tutorials: This image shows the area of a parallelogram

From Trigonometry, it is known that

Opposite side of triangle = Hypotenuse × sin θ

Here, Opposite side of triangle = Height of parallelogram and Hypotenuse = Lateral side of parallelogram. Therefore, we obtain

Area of parallellogram = Base × Lateral side × sin θ

Substituting the Base and Lateral side with the lengths of vectors a and b respectively as shown in the figure below,

Physics Tutorials: This image shows a parrelelogram to help explain vector calculations using trogonometry

we obtain:

|c| = |a| × |b| × sin θ

But we stated before that

c = a × b

and since it is obvious that the above equation is true for magnitudes as well, i.e.

|c| = |a×b|

we obtain

|c| = |a×b| = |a| × |b| × sin θ

What is the direction of the vector product obtained by the cross product of two vectors?

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:

Physics Tutorials: This image shows vectors on a parallelogram

The same idea is used to determine the direction of the vector c which is the cross product vector of a and b. Thus, if we have to find the direction of c = a × b, we start rotating from a to b (in our example this direction is anticlockwise). Therefore, (like in the screw) the direction of the vector c will be upwards as shown in the figure.

Physics Tutorials: This image shows two screws, one being turned anticlockwise and one screw being turned clockwise

On the other hand, if we have to find the cross product d = b × a we must start rotating from b to a. In our example, such direction is clockwise and based on the screwdriver rule, the direction of the vector d will be downwards. Look at the figure:

Physics Tutorials: This image shows anticlocwise vectors

From the two above figures, it is obvious that the vectors c and d are opposite. Hence, we can write

c = -d

Or

a×b = -b× a

Cross product of vectors in coordinates

If the coordinates of the vectors a and b (namely xa, ya, za, xb, yb and zb) are given, we can find the coordinates of the vector c = a × b (i.e. xc, yc and zc) using the following formulae:

xc = ya × zb - yb × za
yc = za × xb - zb × xa
zc = xa × yb - xb × ya

Example

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?

Solution

Before starting the calculation, the position of the two vectors F1 and F2 is as shown below

Physics Tutorials: This image shows clockwise vectors

Thus, the direction of the third fore F3 will be determined by the cross product of the tip's coordinates of the two forces F1 and F2. We have

F3x = F1y × F2z - F1z × F2y
F3y = F1z × F2x - F1x × F2z
F3z = F1x × F2y - F1y × F2x

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

F3x = 3 × (-4) - 0 × 5
= -12 - 0
= -12
F3y = 0 × 1 - (-2) × (-4)
= 0-8
= -8
F3z = (-2) × 5 - 3 × 1
= -10 - 3
= -13

Therefore, the tip of vector F3 will be at (-12, -8, -13). This is illustrated in the figure below. Physics Tutorials: This image shows vectors plotted on a graph

Applications of cross product in Physics

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 M as cross product of Force F and linear distance from the turning point Δx

The formula of Moment of force therefore is:

M = F × Δx

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 F of a conductor at rest as a cross product of Magnetic induction B and the conductor length L multiplied with the scalar I (current).

The formula of Magnetic force therefore is

F = I × (B × L)

3. Magnetic force F of a moving conductor as a cross product of Magnetic induction B and Velocity v multiplied with the scalar q (electric charge).

The formula of Magnetic force in this case is

F = q × (B × v)

4. The angle between two forces F1 and F2 can be calculated using the cross product if the magnitudes of the two vectors F1 and F2 and that of F1 × F2 are known,

and so on.

Example

A conducting wire is placed between the two poles of a horseshoe magnet as shown in the figure. The magnetic field lines (the induction B) 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.

Physics Tutorials: This image shows vectors plotted on a graph with a parallelogram

If the magnitude of the magnetic induction is b = 4 Tesla (B = 4 T) and the amount of electric charges flowing through the wire is q = 6 Coulombs (q = 6 C), find the velocity v of wire (both magnitude and direction) if it forms a right angle to the magnetic field lines. The magnetic force produced is F = 0.2N

.

Solution

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. Physics Tutorials: This image shows a vector of magnetic induction within a horseshoe magnet

The magnitude of velocity is calculated by the equation

F = q × (B × v)

From the cross product rules, we have

|F| = q × |B| × |v| × sin90°

Substituting the known values, we obtain (giving that sin 90° = 1)

0.2N = 6 C × 4 T × |v| × 1
|v| = 0.2N/6 C × 4 T
= 1/120 m/s
= 0.0083 m/s

Mixed product of vectors

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

c = a × b

we can apply the "right hand rule" to find the direction of the vector product c when the directions of a and b are known. Thus, if we put the index and middle fingers of the right hand as shown in the figure,

Physics Tutorials: This image shows a wire being rotated within a horseshoe magnet to illustrate vector calculation of magnetic induction

we obtain the direction of vector

c = a × b

based on the drill (screwdriver) rule. The thumb shows the direction of the product vector.

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