Dot product of two linearly dependant vectors

Question

What does the dot product of two vectors, say $\mathbf a$ and $\mathbf b$, where one is a scaled version of the other represent? If we think in terms of projection then shouldn't the projection equal the length smaller vector? But $ \mathbf a\cdot
\mathbf b $ does not equal $ \mathbf a\cdot \mathbf a $ , if we project the longer to the shorter or shorter to the longer, shouldn't this be true?

Answer 1

Please take into account that the "projection intuition" only works when vector are unit-length. For example, $(\frac{\sqrt2}{2}, \frac{\sqrt2}{2})$ has the same projection on $(1,0)$ than on $(20, 0)$ but yet the dot products with each of them are different. There is a scale-dependency

So, the dot product of two parallel vectors is just the product of their lengths, as can easily be checked. For example, let $(x, y)$ and $(\lambda x, \lambda y)$ be two arbitrary parallel vectors (we'll stick to the 2D-plain, but it's the same story in higher dimensions)

• The product of their lengths is just $||x||\cdot||y|| = \sqrt{x^2+y^2} \sqrt{(\lambda x)^2 + (\lambda y)^2} = \lambda \sqrt{x^2+y^2}^2 = \lambda ||x||^2$
• Their dot product is $\lambda x^2 + \lambda y^2= \lambda ||x||^2$