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Cross Product of Vectors

Let's consider the cross product: an operation between two vectors, u and v, that results in a new vector, w. This can be represented as follows: $$ \vec{u} \ x \ \vec{v} = \vec{w} $$

Understanding the Cross Product

To shed light on this, let's use a practical example.

Imagine you've drawn two vectors on a plane, perhaps using a tool such as Geogebra.

The Outcome of the Cross Product

In such a scenario, the cross product creates a vector that extends perpendicular to the plane, defined by vectors u and v.

Magnitude of the Cross Product

The magnitude of the cross product, represented by u x v, equates to the area of the parallelogram when vectors u and v are considered as its sides.

In this particular instance, the area of the parallelogram is 6.

Computing the Magnitude

This computation accurately reflects the magnitude (or length) of the resulting vector from the cross product, measuring 6 units in length.

Further Aspects of Magnitude

Furthermore, the magnitude of the cross product is equivalent to the product of the magnitudes of vectors |u| and |v|, multiplied by the sine of the angle αuv between vectors u and v. This relationship can be expressed as follows:

$$ |u \ x \ v| = |u| \cdot |v| \cdot \sin \alpha_{uv} $$

Where |v|·sin αuv represents the projection of the second vector.

Identifying the Direction of the Cross Product

The direction of the cross product can be ascertained using the right-hand rule.

Align your right thumb with the first vector (u) and your index finger with the second vector (v) in the cross product, and then fold your middle finger.

right-hand rule

Your middle finger will then indicate the direction of the cross product.

Depending on the specific setup, if you have drawn the vectors on a board, your middle finger will either point towards you or towards the wall.

A Closer Look at the Cross Product

In situations where your middle finger points towards you, the cross product seems to emerge out of the page in your direction.

However, when your middle finger points towards the wall, the cross product appears to recede into the wall.

Take note of this distinction: in the former situation, the cross product is u x v; in the latter, it's v x u. In any cross product operation, your thumb should always align with the first vector. Hence, in the setup on the left, your thumb should correspond with vector u, and in the setup on the right, with vector v.




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