Final answer:
The equation c⟳ =a⟳ b⟳ does not necessarily mean that c = a b without specifying the type of vector multiplication. For a dot product, c equals the product of vectors' magnitudes and the cosine of the angle between them, while for a cross product, it equals the product of vectors' magnitudes and the sine of the angle between them, thus providing different results.
Step-by-step explanation:
The question relates to whether the product of two vectors c⟳ =a⟳ b⟳ implies that their magnitudes multiply to give c = a b. In vector mathematics, there are two main types of multiplication: the dot product (scalar product) and the cross product (vector product). The dot product of two vectors results in a scalar (a number), while the cross product results in another vector. Therefore, the notation c⟳ =a⟳ b⟳ is ambiguous without specifying whether it refers to the dot or cross product.
If c⟳ = a⟳ · b⟳ (dot product), then c is indeed the product of the magnitudes of vectors a and b multiplied by the cosine of the angle between them, hence, c = |a⟳| |b⟳| cos(θ). On the other hand, if c⟳ = a⟳ × b⟳ (cross product), then c⟳ is a vector that is orthogonal to both a⟳ and b⟳ and its magnitude is equal to the area of the parallelogram that a⟳ and b⟳ span, which is c = |a⟳| |b⟳| sin(θ). Hence, without clarity on the type of vector product, one cannot directly conclude that c = a b.