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Physics > System of Particles and Rotational Dynamics > Vector Product of Two Vectors
System of Particles and Rotational Dynamics

Vector Product of Two Vectors

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.

Cross Product

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

c = a × b

So this a × b actually means that the magnitude of c = ab sinθ where θ is the angle between a and b and the direction of c is perpendicular to a well as b.  Now, what should be the direction of this cross product? So to find out the direction, we use the rule which we call it as the ”right-hand thumb rule”.

Suppose we want to find out the direction of a × b here we curl our fingers from the direction of a to b. So if we curl our fingers in a direction as shown in the above figure, your thumb points in the direction of 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 is <180° between a and b. 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 means a × b b × a 

We saw that a × b = c here the thumb is pointing in an upward direction. Whereas in b × a the thumb will point in the downward direction. So, b × a = – c. So it is not commutative.

2] There is no change in the reflection.

What happens to a × b in the reflection? Suppose vector a goes and strikes the mirror, so the direction of a will become – a. So under reflection, a will become – a and b will become – b. Now a × b will become -a × -b =  a × b 

3] It is distributive with respect to vector addition.

This means that if a × ( b + c ) = a × b + a × c. This is true in case of addition.

Vector Product of Unit Vectors

The three unit vectors are \( \hat{i} \) , \( \hat{j} \) and \( \hat{k} \). So,

  • \( \hat{i} \) × \( \hat{i} \) = 0
  • \( \hat{i} \) × \( \hat{j} \) = 1 \( \hat{k} \)
  • \( \hat{i} \) ×\( \hat{k} \) =1 – \( \hat{j} \)
  • \( \hat{j} \) ×\( \hat{i} \) = – \( \hat{k} \)
  • \( \hat{j} \) ×\( \hat{j} \) = 0
  • \( \hat{j} \) ×\( \hat{k} \) = 1 \( \hat{i} \)
  • \( \hat{k} \)× \( \hat{i} \)= \( \hat{j} \)
  • \( \hat{k} \)× \( \hat{j} \)= -\( \hat{i} \)
  • \( \hat{k} \)× \( \hat{k} \)= 0

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

Mathematical Form of Vector Product

a = a\( \hat{i} \) + a\( \hat{j} \) +a\( \hat{k} \)

b = b\( \hat{i} \) + b\( \hat{j} \) +b\( \hat{k} \)

a×b = (  a\( \hat{i} \) + a\( \hat{j} \) +a\( \hat{k} \) ) × ( b\( \hat{i} \) + b\( \hat{j} \) +b\( \hat{k} \) )

= a\( \hat{i} \) × ( b\( \hat{i} \) + b\( \hat{j} \) +b\( \hat{k} \) ) +  a\( \hat{j} \) × ( b\( \hat{i} \) + b\( \hat{j} \) +b\( \hat{k} \) ) + a\( \hat{k} \) × ( b\( \hat{i} \) + b\( \hat{j} \) +b\( \hat{k} \) )

= ab\( \hat{k} \) –  ab\( \hat{j} \) + ab\( \hat{i} \) + ab\( \hat{j} \) – ab\( \hat{i} \)

a×b = (abz  – aby)\( \hat{i} \) + (abx –  abz)\( \hat{j} \) + (ab – abx)\( \hat{k} \)

So the determinant  form of the vectors will be, a×b = 

\( \hat{i} \) \( \hat{j} \) \( \hat{k} \)
ax ay  az
bx by bz

Solved Question

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

  1. Equal to PQ
  2. Less than PQ
  3. Equal to zero
  4. All of the above.

Answer: The correct option is “D”. | \( \vec{P} \)× \( \vec{Q} \) = \( \vec{P} \) \( \vec{Q} \) sinθ, where θ is the angle between P and Q.

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