These are exactly equivalent formulations, merely using different notation for the dot product.

This is exactly equivalent to flipping a coin 100 times to find out what fraction of the time it gives heads or tails.

Experimentally, however we know that the four bonds are exactly equivalent.

Although, in a manner of speaking, the two voyages were exactly equivalent.

In other words, gravitation was exactly equivalent to acceleration.

Table 5.5 presents these coefficients, estimated using ordinary least squares, for exactly equivalent models to those reported in section 5.6.

The size indications are usually similar but not exactly equivalent especially with athletic shoes at extreme sizes.

By contrast, a hard conversion or an adaptive conversion may not be exactly equivalent.

That definition is exactly equivalent to the one above when the values of the random variables are real numbers.

This summarization leads to an approximation that may not be exactly equivalent to the original.