The Limitations Of The Dot Product: When = Then Y=Z Is Not True

if = then y=z in general

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The statement = then y=z is not true in general.

In general, = x1y1 + x2y2 + … + xnyn, where x and y are vectors of length n and their entries are real numbers. Similarly, = x1z1 + x2z2 + … + xnzn.

Therefore, if = , then we have:

x1y1 + x2y2 + … + xnyn = x1z1 + x2z2 + …+ xnzn

Now, we cannot immediately conclude that y = z since the two vectors can have different components but still give the same dot product. For example, let x = (1, 0, 0), y = (1, 1, 1), and z = (1, -1, 0). Then, = 1, = 1, but y is not equal to z.

However, if x is a nonzero vector, then we can conclude that y = z. This is because if x is a nonzero vector, then for any nonzero vector y, we have:

= ||x||*||y||*cos(theta)

where ||x|| is the magnitude of x, ||y|| is the magnitude of y, and theta is the angle between x and y.

Since cos(theta) can only be 1 if theta is 0 (meaning x and y are collinear), we can conclude that if = and x is nonzero, then y and z are collinear, and hence must be equal up to a scalar multiple. That is, y = k*z for some scalar k. Plugging this into the equation = , we get:

||x||*||y|| = ||x||*||z||

Since ||x|| is nonzero, we can cancel it out and obtain:

||y|| = ||z||

This means that the magnitudes of y and z are equal, and since they are collinear, they must be equal up to a scalar multiple. Therefore, y = z.

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