If 𝐀 and 𝐁 are unit vectors, which interval does 𝐀 dot 𝐁, the dot product of 𝐀 and 𝐁, lie in?
Geometrically speaking, the dot product of 𝐀 and 𝐁, 𝐀 dot 𝐁, is the magnitude of 𝐀 times the magnitude of 𝐁 times the cos of the angle 𝜃 between the two vectors. We are told in the question that 𝐀 and 𝐁 are unit vectors. Unit vectors are vectors whose magnitude is one.
And so using this fact, we get that the dot product of 𝐀 and 𝐁 is one times one times cos 𝜃, which is of course just cos 𝜃. So the interval in which the dot product of 𝐀 and 𝐁 lies is the range of values that cos 𝜃 can take.
If we look at a graph of cos 𝜃, we can see that it varies from negative one to one, so negative one is less than or equal to cos 𝜃, which is less than or equal to one. And we’re using less than or equal to signs here not just less than signs because the extreme values of negative one and one are attained by the functions, so they are included in the range of cos.
Another way of writing this is to say that cos 𝜃 is in the interval from negative one to one, where we use the square brackets to get across the fact that both end points, negative one and one, are included in this interval. Cos 𝜃 can take a value of negative one or one.
And so as we discovered that the dot product of 𝐀 and 𝐁 is cos 𝜃, certainly the dot product of 𝐀 and 𝐁 must lie within this interval. Is this our answer? Well first, we need to check that there isn’t any condition on 𝜃.
If 𝜃 were constrained to lie between 60 degrees and 120 degrees, then the cosine of 𝜃, cos 𝜃, would only vary from negative 0.5 to 0.5; actually this would be a smaller interval. But in fact, 𝜃 can take any value from zero degrees up to 180 degrees, and so the dot product of 𝐀 and 𝐁, which is cos 𝜃, can take any value between negative one and one inclusive.