# What can we say if the dot product of two vectors is equal to 1

The question really is in the title. I know what it means if the dot product equals 0 but I find it interesting thinking what it means when it equals exactly 1 and can't seem to find anything online to enlighten me.

Thanks

• Nothing special, really. You get some relation between their lengths and to what degree they point in the same direction, but it's not anything really firm. – Arthur Mar 20 '17 at 23:15
• I don't think it means anything in particular, but check this post for an intuition about the dot product. – Bobson Dugnutt Mar 20 '17 at 23:15
• The value of the dot product has dimensions square length, so it means nothing without a reference pair of lengths to compare it to (namely the lengths of the original vectors) unless it is zero, because this statement does not depend on a choice of units. – Qiaochu Yuan Mar 20 '17 at 23:20
• Thanks very much. I wish I could accept one of these as answers as you have answered my question. Thanks – moony Mar 20 '17 at 23:47

If you already know the vectors are pointing in the same direction, then the dot product equaling one means that the vector lengths are reciprocals of each other (vector b has its length as 1 divided by a's length). For example, 2D vectors of (2, 0) and (0.5, 0) have a dot product of 2 * 0.5 + 0 * 0 which is 1. Also, (1, 1) has a length of sqrt(2), and (0.5, 0.5) has a length of 1/sqrt(2), and the dot product is also 1.
• Your second statement is false, take the vectors $[1,1]$ and $[1/2,1/2]$; the dot product is $1$ but they are not "reciprocals" – Alex Mathers Oct 2 '19 at 1:21