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say you have these two vectors and you just want to know how much of these pointing in the same direction and can I put a number to it so that may not seem like a totally natural question for you yet but come along with me for a few minutes and youll start to see how useful an answer to this question can be and how the dot products solves this problem you didnt know youve had but first we should make this question more precise vectors have two properties length and direction but our question is only about directions so lets forget about the actual length of the vectors for now and replace them with vectors of length 1 that are pointing in the same direction Ill put a hat on them so that we remember that weve done this a good way to interpret this question is how much does one of these vectors overlap with the other what I mean is this you can always write that a hat is equal to some part in the direction of B hat plus a bit thats completely perpendicular to the direction of B wh