Cross Product of Perpendicular Vectors

A vector is a two-dimensional object with both magnitude and direction. A vector can be illustrated geometry as a directed line segment with an arrow pointing in the right direction and a length equal to the vector’s magnitude. The vectors direction is from the tail to the head. At right angles, perpendicular lines cross each other.

In a three-dimensional oriented Euclidean vector space, the cross product or vector product is a binary operation on two vectors. The third vector that is perpendicular to the 2 original vectors is the cross product of two vectors. The area of the parallelogram between them determines its magnitude, and the right-hand thumb rule determines its direction.

Cross Product of Perpendicular Vectors

It is perpendicular to both v and w if the cross product v,w of two nonzero vectors v and w is also a nonzero vector. When two vectors are perpendicular to each other, the angle formed between them is 90 degrees. 

The cross product of two vectors is equal to the product of their magnitudes and the sine of the angle between them, as we all know. The cross product produces a vector that is perpendicular to both vectors because the area vector of any surface is defined in a direction perpendicular to that surface. and whose magnitude equals the area of a parallelogram whose adjacent sides are those two vectors.

                                                                                 Figure 1

If A and B are two independent vectors, the result of their cross product (A×B) is perpendicular to both vectors and normal to the plane in which they are both located. 

A ×B= |A| |B| sin θ n

|A| = length of vector A

B=  length of vector B

θ= angle between A and B

n= unit vector perpendicular to the plane containing a and b

For example, if two vectors are in the X-Y plane, their cross product will result in a resultant vector in the Z-axis’ direction, which is perpendicular to the XY plane. Between the original vectors, the symbol is used. The cross product of two vectors, also known as the vector product, is denoted as:

ab=c

Magnitude of cross product of perpendicular vectors

We get another vector aligned perpendicular to the plane containing the two vectors when we find the cross-product of two vectors. The magnitude of the resultant vector is equal to the product of the sin of the angle between the vectors and the magnitude of the two vectors.  a × b =|a| |b| sin θ.

The magnitude of the cross product is greatest when a and b are perpendicular, as shown by this formula. The cross product, on the other hand, is the zero vector if a and b are parallel or if either vector is the zero vector. There is no unique line perpendicular to both a and b if the vectors are parallel or one vector is the zero vector. Moreover, because there is only one zero-length vector, the cross product is determined uniquely by the definition.

The angle formed by a and c is always 90 degrees.

The angle formed by b and c is always 90 degrees.

We can align a and b parallel to one another or at a 0° angle, resulting in a zero vector.

The original vectors must be perpendicular (angle of 90°) in order for the cross product of the two vectors to be the greatest.

Cross product of perpendicular vectors example

Example: Find the cross-product. Then use the dot product to check perpendicularity.

Solution: 

Now check perpendicularity.

Conclusion

We conclude in this article that when two vectors are perpendicular to each other, the angle between them is 90 degrees. As we all know, the cross product of two vectors equals the product of their magnitudes plus the sine of the angle between them. A vector that is perpendicular to a plane’s basis is also perpendicular to the entire plane. As a result, the cross product of two (linearly independent) vectors is orthogonal to the plane they span because it is orthogonal to each.