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.

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 c 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_x \( \hat{i} \) + a_y \( \hat{j} \) +a_z \( \hat{k} \)

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

a×b = (  a_x \( \hat{i} \) + a_y \( \hat{j} \) +a_z \( \hat{k} \) ) × ( b_x \( \hat{i} \) + b_y \( \hat{j} \) +b_z \( \hat{k} \) )

= a_x \( \hat{i} \) × ( b_x \( \hat{i} \) + b_y \( \hat{j} \) +b_z \( \hat{k} \) ) +  a_y \( \hat{j} \) × ( b_x \( \hat{i} \) + b_y \( \hat{j} \) +b_z \( \hat{k} \) ) + a_z \( \hat{k} \) × ( b_x \( \hat{i} \) + b_y \( \hat{j} \) +b_z \( \hat{k} \) )

= a_x b_y \( \hat{k} \) –  a_x b_z \( \hat{j} \) + a_y b_z \( \hat{i} \) + a_z b_x \( \hat{j} \) – a_z b_y \( \hat{i} \)

a×b = (a_y b_z  – a_z b_y)\( \hat{i} \) + (a_z b_x –  a_x b_z)\( \hat{j} \) + (a_x b_y  – a_y b_x)\( \hat{k} \)

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.