Answer
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Work Step by Step
The dot product, also known as the scalar product, is an operation that takes two vectors and produces a scalar quantity. It is defined as the product of the magnitudes of the two vectors and the cosine of the angle between them.
For two vectors A and B, the dot product is denoted as A · B or A • B.
The dot product satisfies several algebraic laws, including:
1. Commutative Law: A · B = B · A
The dot product of two vectors is commutative, meaning the order of the vectors does not affect the result.
2. Distributive Law: A · (B + C) = A · B + A · C
The dot product distributes over vector addition.
3. Associative Law: (kA) · B = k(A · B) = A · (kB)
The dot product is associative with scalar multiplication.
4. Dot Product with Zero Vector: A · 0 = 0
The dot product of any vector with the zero vector is always zero.
The dot product of two vectors is equal to zero when the vectors are orthogonal, meaning they are perpendicular to each other. In other words, if the angle between two vectors is 90 degrees, their dot product will be zero.
For example, consider two vectors A = (1, 0) and B = (0, 1). The angle between them is 90 degrees, and their dot product is A · B = (1)(0) + (0)(1) = 0.
