 Good morning friends I am Purva and today we will work out the following question. Choose the correct answer and the question is if theta is the angle between two vectors vector A and vector B then vector A dot vector B is greater than equal to 0 only when A theta is greater than 0 and less than pi by 2, B theta is greater than equal to 0 and less than equal to pi by 2, C theta is greater than 0 and less than pi, D theta is greater than equal to 0 and less than equal to pi. Let us now begin with the solution. Now we have to find theta for which vector A dot vector B is greater than equal to 0. Now vector A dot vector B is greater than equal to 0 means vector A dot vector B is equal to 0 and vector A dot vector B is greater than 0. Now first let us consider vector A dot vector B is equal to 0. Now since theta is the angle between vector A and vector B therefore we have cos theta is equal to vector A dot vector B upon mod of vector A into mod of vector B. Now since vector A dot vector B is equal to 0 therefore we get this implies cos theta is equal to 0. Now cos theta is equal to 0 when theta is equal to pi by 2 so we get this implies theta is equal to pi by 2. Now we are also given that vector A dot vector B is greater than 0. Now this implies cos theta is greater than 0 since we know that cos theta is equal to vector A dot vector B upon mod of vector A into mod of vector B. Now cos theta is greater than 0 means cos theta can have all the values which are greater than 0 and the maximum value of cos theta that we can get is 1. So this implies theta is greater than equal to 0 and less than pi by 2 because cos theta will be 1 then theta is equal to 0. Now we mark this as equation 1 and this as equation 2. So from 1 and 2 we get theta is greater than equal to 0 and less than equal to pi by 2 and this is our option B. So we write our answer as B. Hope you have understood the solution. Bye and take care.