Dot product with scalar equal to common symbol for multiplication

Question

I'm in Algebra 2 now, and for about 3 years I have extensively

$$ a \cdot b $$

Instead of

$$ a \times b $$

Now, looking into vector math for shaders, I see this is also the dot symbol is also used to represent dot products. Is my current representation just a form of calculating the dot product with two scalar numbers, which would be the same as multiplication? Or is this a different representation of multiplication used?

score 2 · Accepted Answer · answered Feb 11 '14 at 23:38

2

Note that the field (the collection of scalars) is a vector space over itself. In that case the dot product, and multiplying the scalars is the same thing. It's also the same thing as setting the left one as a scalar and the right as a vector, and multiplying the vector by the scalar.

However it is often the case that for scalars we omit the dot completely and write $ab$.

Also, it is worth pointing out that $\times$, in the $3$-dimensional space is used to denote the cross product, which returns a vector.

answered Feb 11 '14 at 23:38

Asaf Karagila

393,674

Hmmm... so you're saying that $$a \cdot b$$ is actually when a a is scalar and b is a 1D vector? – Liam McInroy Feb 11 '14 at 23:42
You can look at it this way. You can also look at it as a $1$-dimensional dot product of two vectors. Or you can look at it as the product of two scalars in the field. – Asaf Karagila Feb 11 '14 at 23:43
So it is not that it is never incorrect to use either $\cdot$ or $\times$ when working in $1$-dimensional numbers/vectors, as they return the same thing, and are both accepted? – Liam McInroy Feb 11 '14 at 23:45
1

There is a HUGE overload of symbols in mathematics. It's really context dependent. – Asaf Karagila Feb 11 '14 at 23:47
Interesting... Thanks – Liam McInroy Feb 11 '14 at 23:49

Dot product with scalar equal to common symbol for multiplication

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