Final answer:
The dot product of a vector with the cross product it forms with another vector is always zero because the two are perpendicular to each other.
Step-by-step explanation:
The dot product is the result of the scalar multiplication of two vectors and is a scalar referred to as the dot product or scalar product. When it comes to the vector product, also known as the cross product, this results in a vector that is perpendicular to both vectors involved in the multiplication and its magnitude is given by the product of the magnitudes of the two vectors and the sine of the angle between them. If we take a vector A and compute the cross product with another vector B, resulting in a vector A × B, and then compute the dot product of A with A × B, we will have the original vector A dotted with a vector that is perpendicular to it. According to the properties of the dot product, the result of such an operation will always be zero, because the cosine of the angle between A and A × B is zero (since the angle is 90 degrees). Therefore, the answer to the question 'What is the dot product of a vector with the cross product that this vector has with another vector?' is (a) Zero.