Understanding the Relationship Between Vector A and Vector B × Vector C in 3D Space

The question arises when three vectors, A, B, and C, exist and satisfy the relation A.B 0 and A.C 0. This article explores under what conditions the vector A is parallel to the vector cross product B × C. We will delve into the mathematical and geometric interpretation of these conditions to provide a clear and detailed analysis.

Mathematical Background

To begin, let's examine the given conditions:

A.B 0 A.C 0

The dot product of two vectors is given by:

A.B |A| |B| cos(θ)

Where θ is the angle between vectors A and B. Since A.B 0, we can deduce that:

cos(θ) 0

This implies:

θ 90°

Thus, vector A is perpendicular to vector B.

Visualizing Perpendicularity

Considering the perpendicularity condition, let's visualize what happens when A is perpendicular to B. If we place A along the x-axis, and B along the y-axis (the XY plane), it becomes evident that A is aligned perpendicularly to the XY plane. Since A is also perpendicular to C, and C is along the z-axis (the ZY plane), it follows that A must lie along the x-axis.

Vector Cross Product and Perpendicularity

The vector cross product B × C is a vector that is perpendicular to both B and C. In our example:

B is in the y-axis (along the line) C is in the z-axis (up the line)

The cross product B × C will result in a vector that is perpendicular to both B and C. In a Cartesian coordinate system, the vector cross product of any two orthogonal vectors will lie along the remaining axis. Hence, B × C will lie along the x-axis, which is the same direction as vector A.

Conclusion: Parallelism

From the above analysis, we can conclude that vector A is parallel to vector B × C. This is because both vectors lie along the same axis (in our case, the x-axis).

Exceptions and Edge Cases

However, there are edge cases that need to be considered:

When B and C have the same direction, i.e., parallel. In this scenario:

B × C |B||C|sin(0°) 0

The cross product B × C will be zero, as the sine of 0° is zero. Hence, the vector A does not have to be parallel to any vector in this case, since the cross product does not exist.

To better understand the concept, it is helpful to visualize the components using an interactive graph or software tools. By plotting vectors A, B, and C and computing the cross product, one can visually confirm the parallelism of A and B × C.

In summary, the statement "if three vectors A, B, and C satisfy the relation A.B 0 and A.C 0, then the vector A is parallel to vector B × C" is true, except in special cases where B and C are parallel, and the cross product is zero.