Chapter 7.11 - Direction of Cross product

In the previous section, we succeeded in bringing two given vectors $\mathbf\small{\vec{a}}$ and $\mathbf\small{\vec{b}}$ onto the same plane. We also saw the difference between right handed and left handed screws. In this section we will see how to apply the right hand screw rule

For applying the rule, we use the following steps:
1. Place a right handed screw in such a way that, three conditions are satisfied:
(i) The head of the screw is at the tail ends of the two vectors $\mathbf\small{\vec{a}}$ and $\mathbf\small{\vec{b}}$
(ii) The screw is perpendicular to the plane of $\mathbf\small{\vec{a}}$ and $\mathbf\small{\vec{b}}$
(iii) The tip of the screw points towards the positive side of the z-axis
• When these three conditions are satisfied, the screw will be aligned with the magenta line
• This is shown in fig.7.59 (a) below:

Fig.7.59

2. The second step is to turn the screw
There are two possible directions in which the screw can be turned:
(i) The direction shown by the yellow curved arrow in fig.a
(ii) The direction shown by the cyan curved arrow in fig.b
• For finding the direction of ($\mathbf\small{\vec{a}\times \vec{b}}$), we must use the yellow arrow
    ♦ Because, the yellow arrow points from $\mathbf\small{\vec{a}}$ to $\mathbf\small{\vec{b}}$ 
• For finding the direction of ($\mathbf\small{\vec{b}\times \vec{a}}$), we must use the cyan arrow
    ♦ Because, the cyan arrow points from $\mathbf\small{\vec{b}}$ to $\mathbf\small{\vec{a}}$ 
3. The direction of the 'product vector' will be:
The direction of movement of the screw
• Since the screw is right handed:
    ♦ Using the yellow arrow will move the screw upwards. That is., towards the +ve side of z-axis
    ♦ Using the cyan arrow will move the screw downwards. That is., towards the -ve side of z-axis
4. So in our present case:
■ Direction of ($\mathbf\small{\vec{a}\times \vec{b}}$) is towards the +ve side of z-axis
■ Direction of ($\mathbf\small{\vec{b}\times \vec{c}}$) is towards the -ve side of z-axis
Thus:
• If we denote ($\mathbf\small{\vec{a}\times \vec{b}}$) as $\mathbf\small{\vec{c}}$, then, the direction of that $\mathbf\small{\vec{c}}$ will be as shown in fig.7.60(a) below:

Fig.7.60

• If we denote ($\mathbf\small{\vec{b}\times \vec{a}}$) as $\mathbf\small{\vec{c}'}$, then, the direction of that $\mathbf\small{\vec{c}'}$ will be as shown in fig.7.60(b) above

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There is another simpler rule to find the direction. It is called the right hand rule
It can be applied using the following 3 steps:
1. We have seen the significance of the magenta line in the previous section. Using the right hand, grab on the magenta line as shown in fig.7.61(a) below
• The thumb should be in stretched position
• Rest of the four fingers should encircle the magenta line 

Fig.7.61

2. The 'direction in which the four fingers point' gives the following:
• The direction from the 'first vector to be multiplied' to the 'second vector to be multiplied'
• In our present case, the four fingers point from $\mathbf\small{\vec{a}}$ to $\mathbf\small{\vec{b}}$
3. In this situation, the direction of ($\mathbf\small{\vec{a}\times \vec{b}}$) will be given by the 'direction in which the stretched thumb points'

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Note: If we want the direction of ($\mathbf\small{\vec{b}\times \vec{a}}$), we must grab the magenta line as shown in fig.7.61(b). Only then, will the four fingers point from $\mathbf\small{\vec{b}}$ to $\mathbf\small{\vec{a}}$

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So we have completed the discussion on 'direction of the vector product'. In the next section, we will see it's magnitude

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