Scalar Product


I was just wondering why the DOT product is considered to be a scalar. If I take and DOT A with B, then you are saying that we are multiplying the component of vector A along B, directed in the same direction as B, with the Magnitude of Vector B. Now we know that the Product will be in the same direction as B directed along it's unit vector (along it's line of sight); therefore, why do we not assign the product the same unit vector as B??

Thanks for the assistance!


We can calculate the dot product by taking two numbers and multiplying them together. One number is the length of one of the vectors. That is a scalar quantity. The other number is the length of the other vector, a scalar, times the cosine of the angle between them, a scalar, so our second factor, being the product of two scalars is a scalar. Now when we multiply our two factors, both scalar, together the result is the dot, or scalar, product.

It happens that the second factor in our dot product is the length of one vector projected onto the other, but that projection is just a number, not a vector. The scalar nature of the dot product can be confirmed by carrying out the term by term multiplication of the two vectors, just as though they were polynomials, including in each term the unit vector which gives the component its direction. The dot product of two different unit vectors is 0. The dot product of a unit vector with itself is 1. Applying these rules leaves us with a scalar when the multiplication is complete.

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