Let 𝐀 equal negative three, two and 𝐁 equal five, 7.5. Find 𝐀 dot 𝐁.
Here we have these two two-dimensional vectors 𝐀 and 𝐁 and we want to solve for their dot product. In general, given two two-dimensional vectors, and we’ll say that these are 𝐕 one and 𝐕 two, taking their dot product involves multiplying the 𝑥-components and the 𝑦-components separately and then adding these results together. For the vectors 𝐀 and 𝐁 then, their dot product will equal the 𝑥-component of 𝐀 multiplied by the 𝑥-component of 𝐁 plus the 𝑦-component of 𝐀 multiplied by the 𝑦-component of 𝐁. Negative three times five is negative 15, and two times 7.5 is positive 15. So 𝐀 dot 𝐁 is equal to zero. Knowing this, we can move on to the next part of our question.
Which of the following is, therefore, true of the vectors? (A) The two vectors are equal in length. (B) It does not tell anything about the vectors. (C) They are parallel but in opposite directions. (D) They are parallel and in the same direction. (E) They are perpendicular.
So given that 𝐀 dot 𝐁 equals zero, we want to tell which of these five options is true. Whenever we calculate a dot product of two vectors and the result comes out to zero, that has a geometric meaning. That’s because the dot product calculates the overlap between two vectors. So, for example, if our vectors 𝐕 one and 𝐕 two looked like this, then their dot product would depend on this overlap here, that is, the amount that the two vectors lie on top of one another. When the dot product is zero, that means there’s zero overlap. This could be caused by one of the vectors itself being equal to zero.
We see, though, that for 𝐀 and 𝐁 that isn’t the case. These vectors both have a nonzero magnitude. The only other way for this dot product to be zero is if 𝐀 and 𝐁 are perpendicular. In that case, no matter how long these vectors are, their overlap is zero. We see that this answer option is indicated by choice (E). Considering the other options, option (A) says that the two vectors must be equal in length if their dot product is zero. We’ve seen, though, that this need not be the case. A dot product of zero could result from one vector with a zero magnitude being dotted with another nonzero vector.
Option (B) that a dot product of zero doesn’t tell anything about the vectors is also incorrect. A zero dot product does tell us that the two vectors have no overlap. For options (C) and (D), if the two vectors were parallel, that would mean their dot product is not zero. When two vectors are parallel but in opposite directions, as in option (C), their dot product is negative, while a scenario described in option (D) would give a positive dot product. So 𝐀 dot 𝐁 equals zero, and this means the two vectors are perpendicular.