We know that a vector has magnitude as well as a direction. But do we know how any two vectors multiply? Let us now study about the cross product of these vectors in detail.

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## Vector Product of Two Vectors

Vector product also means that it is the cross product of two vectors.

If you have two vectors **a** and **b** then the vector product of **a** and** b** is **c.Â **

**c = aÂ Ã— b**

So thisÂ **aÂ Ã— b** actuallyÂ means that the magnitude of **c = ab sinÎ¸** whereÂ Î¸ is the angle between **a** and** b** and the direction of **c** is perpendicular toÂ **a** well as** b.Â Â **Now, what should be the direction of this cross product? So to find out the direction,Â we use the rule which we call it as the ”right-hand thumb rule”.

Suppose we want to find out the direction ofÂ **aÂ Ã— bÂ **here we curl our fingers from the direction ofÂ **a**Â to**Â b**. So if we curl our fingers in a direction as shown in the above figure, your thumb points in the direction of **cÂ **that is in an upward direction. This thumb denotes the direction of the cross product.

While applying rules to direction, the rotation should be taken to smaller angles that isÂ <180Â° between **a **and** b.Â **So the fingers should always be curled in acute angle between **a **and** b.**

## Properties of Vector Cross Product

1] *Vector product is not commutative.* That meansÂ **aÂ Ã— bÂ **â‰ **bÂ Ã— aÂ **

We saw thatÂ **aÂ Ã— b = cÂ **here the thumb is pointing in an upward direction. Whereas in **bÂ Ã— aÂ **the thumb will point in the downward direction. So,Â **bÂ Ã— a = – c**.**Â **So it is not commutative.

2] *There is no change in the reflection.*

What happens toÂ **aÂ Ã— bÂ **in the reflection? Suppose vector a goes and strikes the mirror, so the direction of **a** will become **– a.Â **So under reflection, **a** will become **– a** and **b** will become **– b**. NowÂ **aÂ Ã— bÂ **will become **-aÂ Ã— -b** =Â Â **aÂ Ã— bÂ **

3]* It is distributive with respect to vector addition.*

This means that if **aÂ Ã— ( b + c )** = **a Ã— b** + **aÂ Ã— c**. This is true in case of addition.

## Vector Product of Unit Vectors

The three unit vectors areÂ \( \hat{i} \) ,Â \( \hat{j} \) andÂ \( \hat{k} \). So,

- \( \hat{i} \)Â Ã—Â \( \hat{i} \) = 0
- \( \hat{i} \)Â Ã—Â \( \hat{j} \) = 1Â \( \hat{k} \)
- \( \hat{i} \)Â Ã—\( \hat{k} \) =1 –Â \( \hat{j} \)
- \( \hat{j} \) Ã—\( \hat{i} \) =Â –Â \( \hat{k} \)
- \( \hat{j} \)Â Ã—\( \hat{j} \) = 0
- \( \hat{j} \) Ã—\( \hat{k} \) = 1Â \( \hat{i} \)
- \( \hat{k} \)Ã—Â \( \hat{i} \)=Â \( \hat{j} \)
- \( \hat{k} \)Ã—Â \( \hat{j} \)= -\( \hat{i} \)
- \( \hat{k} \)Ã—Â \( \hat{k} \)= 0

This is how we determine the vector product of unit vectrors.

## Mathematical Form of Vector Product

a = a_{xÂ }\( \hat{i} \) + a_{yÂ }\( \hat{j} \) +a_{zÂ }\( \hat{k} \)

b = b_{xÂ }\( \hat{i} \) + b_{yÂ }\( \hat{j} \) +b_{zÂ }\( \hat{k} \)

**aÃ—bÂ **= (Â a_{xÂ }\( \hat{i} \) + a_{yÂ }\( \hat{j} \) +a_{zÂ }\( \hat{k} \) )Â Ã— (Â b_{xÂ }\( \hat{i} \) + b_{yÂ }\( \hat{j} \) +b_{zÂ }\( \hat{k} \) )

= a_{xÂ }\( \hat{i} \)Â Ã— (Â b_{xÂ }\( \hat{i} \) + b_{yÂ }\( \hat{j} \) +b_{zÂ }\( \hat{k} \) ) +Â a_{yÂ }\( \hat{j} \)Â Ã—Â (Â b_{xÂ }\( \hat{i} \) + b_{yÂ }\( \hat{j} \) +b_{zÂ }\( \hat{k} \) ) +Â a_{zÂ }\( \hat{k} \)Â Ã—Â (Â b_{xÂ }\( \hat{i} \) + b_{yÂ }\( \hat{j} \) +b_{zÂ }\( \hat{k} \) )

=Â a_{xÂ }b_{yÂ }\( \hat{k} \) –Â a_{xÂ }b_{zÂ }\( \hat{j} \) + a_{yÂ }b_{zÂ }\( \hat{i} \) +Â a_{zÂ }b_{xÂ }\( \hat{j} \) – a_{zÂ }b_{yÂ }\( \hat{i} \)

**aÃ—bÂ **= (a_{yÂ }b_{z }Â – a_{zÂ }b_{y})\( \hat{i} \)Â + (a_{zÂ }b_{x}Â –Â a_{xÂ }b_{z})\( \hat{j} \) + (a_{xÂ }b_{yÂ }Â – a_{yÂ }b_{x})\( \hat{k} \)

So the determinantÂ form of the vectors will be,Â **aÃ—b =Â **

\( \hat{i} \) | \( \hat{j} \) | \( \hat{k} \) |

a_{x} |
a_{y} |
Â a_{z} |

b_{x} |
b_{y} |
b_{z} |

## Solved Question

Q1.Â The magnitude of the vector product of two vectors \( \vec{P} \) and \( \vec{Q} \) may be:

- Equal to PQ
- Less than PQ
- Equal to zero
- All of the above.

Answer: The correct option is “D”. |Â \( \vec{P} \)Ã— \( \vec{Q} \) =Â \( \vec{P} \) \( \vec{Q} \) sinÎ¸, whereÂ Î¸ is the angle between P and Q.

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