Determining Vector B Given Scalar and Vector Products with Nonzero Vector A

In this article, we will explore the conditions under which the scalar product and vector product of a given nonzero vector A and an unknown vector B are both zero. This analysis will provide insight into the nature of vector B, leading us to a definitive conclusion about its properties.

Introduction

Suppose you are given a nonzero vector ( mathbf{A} ) and an unknown vector ( mathbf{B} ). The problem presents two conditions: (1) the scalar product (dot product) of ( mathbf{A} ) and ( mathbf{B} ) is zero, and (2) the vector product (cross product) of ( mathbf{A} ) and ( mathbf{B} ) is also zero. These conditions can be mathematically represented as:

Scalar Product (Dot Product): ( mathbf{A} cdot mathbf{B} 0 ) Vector Product (Cross Product): ( mathbf{A} times mathbf{B} mathbf{0} )

Analysis of the Conditions

We will examine each of these conditions in detail and determine what can be concluded about the vector ( mathbf{B} ).

Scalar Product (Dot Product)

The scalar product (dot product) of two vectors is defined as:

Let ( mathbf{A} cdot mathbf{B} |mathbf{A}| |mathbf{B}| cos theta )

where ( |mathbf{A}| ) and ( |mathbf{B}| ) are the magnitudes of vectors ( mathbf{A} ) and ( mathbf{B} ), and ( theta ) is the angle between them. Given the condition ( mathbf{A} cdot mathbf{B} 0 ), we have:

If ( mathbf{B} ) is the zero vector, then ( |mathbf{B}| 0 ), and thus the dot product is always zero regardless of the direction of ( mathbf{A} ). If ( mathbf{B} ) is not the zero vector, for the dot product to be zero, the angle ( theta ) between them must be either 90 degrees (perpendicular) or 180 degrees (opposite direction), or the magnitude of ( mathbf{B} ) must be zero, which is not possible as ( mathbf{B} ) is an unknown vector.

Therefore, the most consistent and feasible conclusion for the scalar product condition is that ( mathbf{B} ) must be the zero vector.

Vector Product (Cross Product)

The vector product (cross product) of two vectors is defined as:

Let ( mathbf{A} times mathbf{B} |mathbf{A}| |mathbf{B}| sin theta mathbf{n} )

where ( |mathbf{A}| ) and ( |mathbf{B}| ) are the magnitudes of vectors ( mathbf{A} ) and ( mathbf{B} ), ( theta ) is the angle between them, and ( mathbf{n} ) is a vector normal to the plane containing ( mathbf{A} ) and ( mathbf{B} ). Given the condition ( mathbf{A} times mathbf{B} mathbf{0} ), we have:

If ( mathbf{B} ) is the zero vector, then ( |mathbf{B}| 0 ), and thus the cross product is always zero regardless of the direction of ( mathbf{A} ). If ( mathbf{B} ) is not the zero vector, for the cross product to be zero, either ( |mathbf{A}| 0 ) or ( theta 0 ) or ( theta 180 ) degrees (parallel or antiparallel). Since ( mathbf{A} ) is given as a nonzero vector, the only feasible solution is that ( mathbf{B} ) must be the zero vector.

Thus, the vector product condition also supports the conclusion that ( mathbf{B} ) must be the zero vector.

Conclusion

From the analysis of both the scalar product and vector product conditions, we can conclude that the vector ( mathbf{B} ) must indeed be the zero vector. This conclusion is consistent with the properties of the zero vector in vector space.

Additional Considerations

It is important to note that there is no such thing as an “unknown zero vector.” Every vector space has exactly one zero vector, which is the vector with all components equal to zero. Therefore, the conditions described in the problem are uniquely satisfied when ( mathbf{B} ) is the zero vector.

Key Points

The scalar product of ( mathbf{A} ) and ( mathbf{B} ) being zero implies that either ( mathbf{B} ) is the zero vector or ( mathbf{A} ) and ( mathbf{B} ) are perpendicular. The vector product of ( mathbf{A} ) and ( mathbf{B} ) being zero implies that either ( mathbf{B} ) is the zero vector or ( mathbf{A} ) and ( mathbf{B} ) are parallel or anti-parallel, but given that ( mathbf{A} ) is nonzero, ( mathbf{B} ) must be the zero vector. The uniqueness of the zero vector in vector space ensures that these conditions lead to the same conclusion.

In summary, the analysis of the scalar and vector products of a nonzero vector ( mathbf{A} ) with an unknown vector ( mathbf{B} ) leads to the conclusion that ( mathbf{B} ) must be the zero vector.

For further reading or discussion on vector products and dot products in vector spaces, consider the following references:

Dot Product - Wikipedia Cross Product - Wikipedia Dot Product (Scalar Product) and Norm - LibreTexts