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autodidude
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Is there a way of deriving the algebraic definition of the dot product from the geometric definition without using the law of cosines?
Strictly speaking the term "dot product" is only used Euclidean space. In other vector spaces the term is "inner product". Of course, any n-dimensional vector space is isomorphic to Rn so the two work out to be "essentially" the same.jedishrfu said:Also one thing to be aware of is that the algebraic defintion for vector dot and cross products only work when you have your vectors defined in a Euclidean space like our old favorite x,y,z or i,j,k.
and of course here's more info on it from wikipedia:
http://en.wikipedia.org/wiki/Vector_dot_product
autodidude said:Is there a way of deriving the algebraic definition of the dot product from the geometric definition without using the law of cosines?
The dot product, also known as the scalar product, is a mathematical operation that takes two vectors as input and produces a scalar value as output. The algebraic definition of the dot product is the sum of the products of the corresponding components of the two vectors. In other words, it is the multiplication of the first component of the first vector with the first component of the second vector, added to the multiplication of the second component of the first vector with the second component of the second vector, and so on.
To calculate the dot product of two vectors, you first multiply the corresponding components of the two vectors, and then add all of those products together. For example, if you have two vectors, A = (a1, a2, a3) and B = (b1, b2, b3), the dot product can be calculated as a1b1 + a2b2 + a3b3.
The dot product has several important applications in mathematics and physics. It can be used to calculate the angle between two vectors, determine if two vectors are perpendicular, and project a vector onto another vector. It also has applications in fields such as computer graphics, engineering, and economics.
The dot product is related to the magnitude and direction of vectors in several ways. For example, if the dot product of two vectors is zero, it means that the vectors are perpendicular to each other. If the dot product is positive, it means that the vectors are pointing in the same general direction, and if it is negative, they are pointing in opposite directions. The magnitude of the dot product also gives information about the length of the projection of one vector onto another.
The dot product has several properties, including commutativity (A · B = B · A), distributivity (A · (B + C) = A · B + A · C), and associativity with scalar multiplication (k(A · B) = (kA) · B = A · (kB)). It is also related to the magnitude of vectors, as the dot product of a vector with itself is equal to the square of its magnitude. Additionally, the dot product is used to define the concept of orthogonality, where two vectors with a dot product of zero are considered orthogonal or perpendicular to each other.