# Projections vs Dot Products Consider the difference between [Projections](Projection.md) and [Dot Products](Dot%20Product.md). A projection as a very clear geometric interpretation, as shown below. But what exactly is the dot product, and how do these two relate?  The answer is actually quite straight forward. A projection is an asymmetric operation, meaning projecting $x$ onto $v$ does not yield the same vector as projecting $v$ onto $x$: $\text{proj}_v(x) \ne \text{proj}_x(v)$ This is because projection takes the vector that we are projecting *onto* and only focuses on it's direction—it discards any information about it's magnitude. However, it *keeps* the magnitude information about the vector that is being projected. Thus, depending on which vector is being projected onto and which is being projected, we'll get a different result. This is fine, but if we are trying to determine the [Similarity](Similarity-Measures.md) of two vectors, we'd like to come up with a measure that is invariant to the role that the two vectors play. We could just focus on the *angle* between the two vectors, constraining our focus entirely to how similar their direction is. This is exactly what **cosine similarity** does. But what if we also want to account for *magnitude*? After all, it may be that if $x$ and $v$ point in the same directions, but $x$ has a magnitude that is 100 times greater than $v$, we need to know about it. This is exactly where the dot product comes in. It is sensitive to both direction *and* magnitude, and it is *symmetric* --- Date: 20250620 Links to: Tags: References: * []()