Unveiling Perpendicularity
3. Using the Dot Product to Check for Right Angles
Perpendicular vectors, meeting at a perfect 90-degree angle, have a unique relationship revealed through something called the dot product. The dot product is an operation that takes two vectors and returns a single number (a scalar). The magic happens when the dot product equals zero. If the dot product of two vectors is zero, those vectors are perpendicular!
So, how do you calculate the dot product? For two-dimensional vectors a = (a1, a2) and b = (b1, b2), the dot product is calculated as: a b = (a1 b1) + (a2 b2). For three-dimensional vectors a = (a1, a2, a3) and b = (b1, b2, b3), it's: a b = (a1 b1) + (a2 b2) + (a3 b3). You simply multiply corresponding components and then add the results.
Let's try an example. Say vector a = (2, -1) and vector b = (1, 2). The dot product is (2 1) + (-1 2) = 2 - 2 = 0. Voila! The dot product is zero, so a and b are perpendicular.
What if the dot product isn't zero? Let's say vector a = (1, 1) and vector b = (2, 1). The dot product is (1 2) + (1 1) = 2 + 1 = 3. Since the dot product is 3 (not zero), these vectors are not perpendicular. They intersect, but not at a right angle.
Neither Parallel Nor Perpendicular: The In-Between Space
4. Identifying Vectors That Don't Fit the Mold
So, what happens when vectors are neither parallel nor perpendicular? Well, that's actually the most common scenario! In this case, the vectors simply intersect at some angle other than 0, 90, or 180 degrees. They're just hanging out, pointing in their own directions, not following any specific rules.
Remember those vectors from the previous example where vector a = (1, 1) and vector b = (2, 1)? We already determined that their dot product wasn't zero, so they're not perpendicular. Also, there's no scalar you can multiply a by to get b . Therefore, they're not parallel either. They fall into the "neither" category.
Essentially, to check if two vectors are in this "neither" category, you just need to rule out parallel and perpendicular. If they don't satisfy the conditions for either of those relationships, they automatically fall into this third category.
Don't feel bad if your vectors end up in the "neither" category. It doesn't mean they're useless! They still have magnitude and direction, and they can be used in all sorts of vector operations. They just don't have the special relationship of being perfectly aligned or perfectly orthogonal.
