Final answer:
The true or false statements about vector (cross product) and scalar (dot product) products for non-zero vectors A and B involve understanding that these operations can result in zero or non-zero values depending on the vectors' relations, such as being parallel, antiparallel, or orthogonal.
Step-by-step explanation:
The question pertains to the properties of vector product (cross product) and scalar product (dot product) for non-zero vectors A and B. Analyzing the statements given:
- a) True, the cross product of A and B is always zero. - This statement is false. The cross product of two non-zero vectors A and B (A × B) is zero only if A and B are parallel or antiparallel. Otherwise, it is non-zero and is a vector perpendicular to both A and B.
- b) False, the dot product of A and B is always zero. - This statement is true as it correctly asserts that the dot product is not always zero. The dot product is zero only when A and B are orthogonal (perpendicular to each other).
- c) True, the dot product of A and B is always non-zero. - This statement is false. The dot product of two non-zero vectors can be zero if the vectors are orthogonal to each other.
- d) False, the cross product of A and B is always non-zero. - This statement is true because the cross product of two non-parallel, non-zero vectors is always a non-zero vector, which is perpendicular to the plane containing vectors A and B.
Both the dot product and the cross product have different properties and result in different mathematical objects. The dot product produces a scalar, which depends on the cosine of the angle between two vectors, while the cross product produces a vector, which depends on the sine of the angle and is anticommutative.