What is a cross product?

The cross product of two vectorsA vector is a quantity that has a magnitude and a direction. is a mathematical operation that results in a vector, perpendicularTwo vectors are perpendicular to each other if they intersect at 90 degrees, simply the angle between them is 90. to the two original vectors we take the cross product of. It is used in various fields, including physics, mathematics, and computer graphics, to understand and deal with vector behaviors in a 3D space.

Suppose that we have two vectors: $v$ and $w$. We represent their cross product (pronounced as $v$ cross $w$) as:

• $u$ : The resulting vector that is always perpendicular to both $v$ and $w$.

• $v$ and $w$: The two vectors which are linearly independent and are taken cross product of.

Note: Linearly independent vectors cannot be expressed as a linear combination of each other meaning both vectors don't have the same direction.

Representation

The cross product can be represented in two ways: geometric and algebraic.

Geometric representation

Geometrically, the formula to find the cross product of the two vectors is:

• $u$ : The resultant vector.

• $|v|$ : The magnitudeLength or size of the vector. of the vector $v$.

• $|w|$: The magnitude of the vector $w$.

• $θ$: The angle between vectors $v$ and $w$.

• $n$ : The direction of the resultant vector $u$.

The magnitude of the resultant vector

The magnitude of the resultant vector is important in various applications where understanding the area or strength of a vector's effect is important. For instance, in physics, when we find the torque using the cross product, the magnitude helps us understand the strength of our resultant vector's force. We can find the magnitude by calculating the area of a parallelogram that is formed by $v$ and $w$.

• $base = |v|$

• $height = |w|sinθ$

The direction of the resultant vector

When we calculate the cross product of two vectors, our resultant vector has two possible directions: one towards the negative axis and the other towards the positive. This direction cannot be calculated directly through the above formula, so we use the right-hand rule to find it.

Right hand rule

We follow the following instructions in the right hand rule trick.

1. We use our right hand.

2. We point our index finger in the direction of the first vector, $v$.

3. We point our middle finger in the direction of the second vector, $w$.

4. The resultant vector, $u$, will point in the direction of our extended thumb.

Example

Let's understand the use of the following formula using an example. We can calculate the magnitude of our resultant vector $u$ if we skip the direction in the formula.

Let's say $v = (2, 3, 1)$ and $w = (4, 2, 5)$. We suppose both the vectors are perpendicular to each other so the angle between them will be $90$.

Magnitude of $v$:

Magnitude of $w$:

Value of $sinθ$:

Magnitude of $u$:

Note: Our cross product is maximum when the vectors are perpendicular (angle equals $90$) and minimum when the vectors are parallel (angle equals $0$).

Algebraic representation

Algebraically, the cross product between two vectors is calculated using a matrix determinant approach. To understand the concept of the determinant of a matrix, we can refer to this answer.

Let's suppose we have $v$ and $w$ as two vectors such that:

$v = (v1, v2, v3)$

$w = (w1, w2, w3)$

The cross product will be:

• Here i, j, and k are unit vectors in the x, y, and z directions, respectively.

We get the following formula after simplifying the determinant form:

Example

Let's say $v = (2, 3, 1)$ and $w = (4, 2, 5)$

The cross product will be:

Properties

Let's discuss the properties of a cross product. Let's suppose $v$, $w$ and $u$ are three vectors and $a$ is a scalar.

Conclusion

The cross product of vectors is a fundamental mathematical operation with wide-ranging applications in various fields. By taking the cross product of two vectors, we obtain a new vector perpendicular to both the original vectors.