Lecture Note
University
Massachusetts Institute of TechnologyCourse
Multivariable CalculusPages
2
Academic year
2023
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26
Perpendicular vectors ● Perpendicular vectors are two or more lines that intersect each other at right angles. These are also called orthogonal vectors. The angle between the two lines is 90degrees. ● Perpendicular vectors are two vectors whose dot product is zero. A vector that is perpendicular to another vector has a 90 degree angle with it. The two vectors areequal in magnitude but opposite in direction. The thing is that if we know the vectors v and w and want to find the angle between them,this formula can be used because it is easy to determine. It is straightforward to find the vector v dot w, and it's also easy to find the norms of v andw. We can use this formula to find the cosine of theta. The dot product allows us to find the angle between two vectors, and a related thing is thatwe can determine when two vectors are perpendicular. This we call perpendicular vectors. If v is perpendicular to w, then the dot product of v and w is equal to the square of themagnitude of v times the square of the magnitude of w times the cosine of their angle,which is twice pi over two. The cosine of twice pi over two is zero. The dot product of 0 and v is 0, so v is perpendicular to w. It is checkable by taking the dot product. v dot w is 0.
One useful approach is to find a vector that points in the same direction as some existingvector. Another thing we often want to do is find a vector that's perpendicular to a vectorwe already know. It is possible to do in two different ways. If we want to find a vector that's in the same direction as a known vector, we can see thefollowing. The same direction as 2,1 is 2. It means lambda times 2.1. And to find a vector that's perpendicular to one we already know, here what we can think. vand w are perpendicular if their dot product is zero. Given a vector v, we can find another vector which is perpendicular to v by performing thedot product between v and 2,3. We have the equation 0 is v dotted with 2,3, which is 2v1plus 3v2. We can then solve the equation by finding numbers v1 and v2. It should not be challengingfor you.
Perpendicular Vectors
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