I am going through MIT's course on Multivariable calculus Given by Herbert Gross. In the Dot product section of the question is as follows : With A and B as in part (a) use the result of part (b) to interpret geometrically the set of all vectors C for which A • B = A • C, where we proved in part B that A is perpendicular to (B-C).
Answer in the book is : Since we have already agreed that vectors do not depend on their point of origin, let us assume, without loss of generality, that and B originate at a common point, the origin. Now, the locus of all vectors V such that A • V = 0 is the plane to which A is perpendicular. (Again, for the sake of uniformity, we -f are assuming that all vectors V start at the origin.) A picture is provided.
Q1: Why can I bring any vector back to the Origin (0,0,0) ? Q2: How is it that the set of all vectors perpendicular to A (called V) is a Plane. I thought to get a plane you need two independent vector to span a plan, there is no mention of that here.
I was thinking simply that all this means is that if i have any vectors B and C then their difference B-C is always perpendicular to A.
Q3: What is the purpose of all this construction of plane and point OP and what not ?
A1: In Euclidean geometry, translation unambiguously identifies vectors based at different points. In fancy terms, parallel transport is path-independent.
A2: We do need two vectors to span a plane, but (in three-dimensional Euclidean geometry) a non-zero vector $\vec{A}$ is normal to a plane, really, to a unique one-parameter family of pairwise disjoint, mutually-parallel planes.
A3: The purposes of this construction are too numerous to summarize. As explanation for the comment: