Vector product (cross product)

Understanding the Concept

The vector product, also known as the cross product, of two vectors results in a third vector.
Unlike the scalar (dot) product, the vector product is not commutative, meaning u x v is not the same as v x u.
The vector product of two vectors is a vector that is perpendicular to the plane containing the original vectors.
The magnitude (length) of the resulting vector is equal to the product of the magnitudes of the two vectors and the sine of the angle between them.

Calculating the Vector Product

To calculate the cross product, use the determinant of a 3x3 matrix consisting of unit vectors in the first row, components of the first vector in the second row and components of the second vector in the third row.
It’s crucial to remember that the vector product produces a vector, not a scalar.
The cross product of vectors will be zero if the vectors are parallel to each other, because the angle between them is 0 degrees and sine of 0 is zero.

Common Applications of Vector Product

The vector product can be used to calculate the area of a parallelogram if the magnitudes and angle of two vectors forming the parallelogram are known.
The cross product also plays a pivotal role in physics, often used for concepts like torque and angular momentum.

Potential Pitfalls

Remember that the order in which vectors are crossed matters; u x v = - (v x u).
Not all vectors can be crossed. Make sure the vectors are three dimensional; the vector product is not defined for two dimensional vectors.
Be careful when evaluating the determinant of the 3x3 matrix; it’s easy to mix up the signs.

Remember that vectors are crucial elements in further mathematics. Therefore, understanding the concept and applications of vector products can lead you to decipher more complex problems.