Understanding Scalar and Vector Products of Vectors A and B

When working with vectors, understanding the scalar (dot) product and vector (cross) product is crucial for determining the relationship between two vectors. In this discussion, we explore the conditions under which a nonzero vector A has a scalar product of zero with an unknown vector B, and a nonzero vector product with B. We will delve into the implications of these conditions and provide numerical examples to support our conclusions.

Scalar Product of Vectors

The scalar product (or dot product) of two vectors, denoted as A · B, is defined as the product of their magnitudes and the cosine of the angle between them. Mathematically, this is expressed as:

A · B |A| |B| cos θ

Where |A| and |B| are the magnitudes of vectors A and B, and θ is the angle between them. If the scalar product of vectors A and B is zero, this implies either that at least one of the vectors is the zero vector or that the two vectors are perpendicular to each other.

Cases for Zero Scalar Product

Let's consider the case where the scalar product of vectors A and B is zero:

One of the vectors is the zero vector: If either A or B is the zero vector, then the scalar product will be zero. The vectors are perpendicular: If the angle θ between vectors A and B is 90 degrees, then cos θ 0, and the scalar product will be zero.

Numerical Examples

Let's look at a numerical example to illustrate these principles. Suppose A 100i 0j 0k and B 0i 100j 0k. In this case:

The scalar product A · B (100*0) (0*100) (0*0) 0 The vector product A × B (100i 0j 0k) × (0i 100j 0k) 0i - 0j (100 * 100 - 0 * 0)k 10000k, which is nonzero.

Here, the vectors A and B are perpendicular, and the scalar product is zero, as expected.

Vector Product of Vectors

The vector product (or cross product) of two vectors, denoted as A × B, is a vector that is perpendicular to both A and B. The magnitude of the vector product is given by:

|A × B| |A| |B| sin θ

By definition, the vector product of two vectors is nonzero unless the vectors are parallel (i.e., the angle between them is 0 or 180 degrees). If the vector product of A and B is zero, this implies that the vectors are parallel.

Let's use the same vectors from the scalar product example: A 100i 0j 0k and B 0i 100j 0k. In this case, the vector product is 10000k, which is nonzero, as expected.

Conclusion

In summary, if the scalar product of a nonzero vector A with an unknown vector B is zero, then B must either be the zero vector or parallel or perpendicular to A. If the vector product of A and B is nonzero, then they are not parallel.

To apply these concepts, numerical examples can be used to verify the conditions and understand the relationships between vectors. For instance, if A 100 and B 34, the scalar product can be calculated as 100 * 34 * cos(θ). If cos(θ) 0, then the scalar product is zero, and the vectors are perpendicular.

It is important to note that there is no such thing as an “unknown zero vector.” Every vector space contains exactly one zero vector, which is the vector with all components equal to zero.