Last time I have written about the dot product of unit vectors that is:

i.i=1

j.j=1

k.k=1

i.j=j.k=k.i=0

Today I will discuss about the cross product of unit vectors:

The cross product of unit vectors i, j and k is an important topic in vectors and is frequently used in Physics and Mathematics.

Many students find questions such as

i × j = ?
j × k = ?
k × i = ?

confusing at first. The good news is that these results follow a simple pattern.

In this article, we will learn the cross product of unit vectors in a simple way, including the positive and negative results, the right-hand rule, examples and common mistakes.

What are i, j and k?

In three-dimensional Cartesian coordinates, i, j and k are unit vectors along the three coordinate axes.

i is the unit vector along the x-axis

j is the unit vector along the y-axis

k is the unit vector along the z-axis

A unit vector has a magnitude of 1.

So,

|i| = |j| = |k| = 1

You can think of them as three mutually perpendicular directions:

What is the cross product?

The cross product, also called the vector product, is written using the symbol ×.

For two vectors A and B,

|A × B| = AB sin θ

where θ is the angle between the two vectors.

The result of a cross product is another vector.

Its direction is perpendicular to both of the original vectors.

For example, when two perpendicular vectors are crossed,

θ = 90°

and therefore,

sin 90° = 1

So their cross product has maximum magnitude.

Cross Product of i and j

The unit vectors i and j are perpendicular to each other.

Using the right-hand rule, the direction perpendicular to both the x-axis and y-axis is the positive z-direction.

The positive z-direction is represented by k.

Therefore,

i × j = k

This is one of the three most important results to remember.

Cross Product of j and k

Now consider j × k.

The vectors j and k represent the positive y- and z-directions.

Using the same cyclic pattern, the result is in the positive x-direction.

Therefore,

j × k = i

Cross Product of k and i

Similarly, for k × i, the result is in the positive y-direction.

Therefore,

k × i = j

So the three important results are:

Cross product	Result
i × j	k
j × k	i
k × i	j

Easy way to remember

Remember the cycle:

i → j → k → i

When the vectors are taken in this order, the answer is positive.

So:

i × j = k

j × k = i

k × i = j

What happens when the order is reversed?

The cross product is not commutative.

This means that changing the order changes the sign.

The important rule is:

A × B = −(B × A)

For example, we know:

i × j = k

Therefore,

j × i = −k

Similarly,

j × k = i

so,

k × j = −i

And,

k × i = j

so,

i × k = −j

The complete table is:

Cross product	Result
i × j	k
j × k	i
k × i	j
j × i	−k
k × j	−i
i × k	−j

Simple memory trick

Think of i → j → k → i as the forward direction.

Forward order → positive

Reverse order → negative

For example:

i × j = +k

but

j × i = −k

What is i × i?

What happens when a unit vector is crossed with itself?

For example,

i × i

The angle between i and i is 0°.

Using the cross-product formula,

|i × i| = |i||i| sin 0°

Since

sin 0° = 0

we get:

i × i = 0

Similarly,

j × j = 0

and

k × k = 0

So:

Same unit vector × same unit vector = 0

This gives another easy rule:

Same → 0

The easiest way to remember all the results

You only need to remember three ideas:

1. Forward order → positive

i × j = k

j × k = i

k × i = j

2. Reverse order → negative

j × i = −k

k × j = −i

i × k = −j

3. Same vector → zero

i × i = 0

j × j = 0

k × k = 0

This is much easier than trying to memorize all nine results separately.

Right-Hand Rule

The right-hand rule helps us find the direction of a cross product.

To find the direction of A × B:

1. Point the fingers of your right hand in the direction of A.
2. Curl your fingers toward B.
3. Your thumb points in the direction of A × B.

For example, for

i × j

the right-hand rule gives the positive z-direction.

Therefore,

i × j = k

This is why the direction of the cross product is important. The answer is not just a number; it is a vector with a direction.

Two Simple Examples

Numerical Example 1: Cross Product of Two Vectors

Question:
Find the cross product of the following two vectors:

A = 3i + 2j

B = 4j + 5k

Solution:

Using the distributive property of the cross product:

A × B = (3i + 2j) × (4j + 5k)

Expanding the expression:

A × B = 12(i × j) + 15(i × k) + 8(j × j) + 10(j × k)

We know that:

i × j = k
i × k = −j
j × j = 0
j × k = i

Therefore:

A × B = 12k − 15j + 0 + 10i

Rearranging the terms:

A × B = 10i − 15j + 12k

Therefore, the cross product is:

Answer: A × B = 10i − 15j + 12k

Numerical Example 2: Simple Cross Product

Question:
Find the cross product of the following vectors:

A = 2i

B = 5j

Solution:

A × B = (2i) × (5j)

Taking the numerical coefficients outside:

A × B = 10(i × j)

We know that:

i × j = k

Therefore:

A × B = 10k

Hence, the cross product is:

Answer: A × B = 10k

What happens if the order is reversed?

B × A = (5j) × (2i)

B × A = 10(j × i)

Since:

j × i = −k

Therefore:

B × A = −10k

This shows that the cross product is not commutative. In other words:

A × B ≠ B × A

Cross Product vs Dot Product

Students often confuse the cross product and dot product.

The symbols are different:

Dot product → A · B

Cross product → A × B

The results are also different.

Dot Product	Cross Product
Uses ·	Uses ×
Result is a scalar	Result is a vector
A · B = AB cos θ	A × B = AB sin θ
Zero for perpendicular vectors	Maximum for perpendicular vectors

For example:

i · j = 0

because i and j are perpendicular.

But,

i × j = k

because their cross product gives a vector perpendicular to both.

Where is Cross Product Used in Physics?

The cross product is not just a mathematical rule. It is used in several important areas of Physics.

1. Torque

Torque is given by:

τ = r × F

where r is the position vector and F is the force.

2. Magnetic Force

The magnetic force on a moving charged particle is given by:

F = q(v × B)

where v is velocity and B is the magnetic field.

These formulas show why understanding the direction of a cross product is important in Physics.

Common Mistakes Students Make

Mistake 1: Writing i × j = −k

The correct result is:

i × j = k

The negative sign appears when the order is reversed:

j × i = −k

Mistake 2: Thinking cross product is commutative

It is not.

In general,

A × B ≠ B × A

Instead,

A × B = −B × A

Mistake 3: Thinking i × i = i

The cross product of a vector with itself is zero:

i × i = 0

Mistake 4: Confusing dot and cross products

Remember:

i · j = 0

but

i × j = k

The dot product gives a scalar, while the cross product gives a vector.

Quick Revision Table

Expression	Answer
i × j	k
j × k	i
k × i	j
j × i	−k
k × j	−i
i × k	−j
i × i	0
j × j	0
k × k	0

One-line memory trick

i → j → k → i = positive

Reverse order = negative

Same vector = zero

Frequently Asked Questions

What is i × j?

The cross product of i and j is:

i × j = k

What is j × k?

j × k = i

What is k × i?

k × i = j

What is j × i?

Since the order is reversed,

j × i = −k

What is i × i?

i × i = 0

The same is true for j × j and k × k.

Why does the cross product have a direction?

The cross product produces a vector that is perpendicular to both of the original vectors. Its direction can be determined using the right-hand rule.

Is i × j the same as j × i?

No.

i × j = k

whereas

j × i = −k

Changing the order changes the sign.

Conclusion

The cross product of unit vectors becomes easy when you remember a simple pattern:

i → j → k → i

The forward order gives positive results:

i × j = k

j × k = i

k × i = j

Reversing the order changes the sign:

j × i = −k

k × j = −i

i × k = −j

And crossing a unit vector with itself gives zero:

i × i = j × j = k × k = 0

Once this pattern is understood, most basic questions on the cross product of unit vectors can be solved quickly. I hope that now you can understand and explain everything about the cross or vector product of two unit vectors. You can learn about dot products here.