Finding Two Orthogonal Vectors in R3Date: 12/15/2003 at 21:03:14 From: Emerson Subject: Find two non zero vectors in R3 .......... I'm supposed to find two non-zero vectors in R3 that are orthogonal, but I'm pretty confused about the whole idea. Can you explain it to me? Date: 12/25/2003 at 20:15:46 From: Doctor Jordan Subject: Re: Find two non zero vectors in R3 .......... Dear Emerson, R3 is three dimensional space; that is, it contains vectors with three entries. Therefore, [1] is not in R3, while [1] is in R3. [1] [1] [1] [1] [1] (Note that there are different R3s; in R4 there are infinitely many different R3s. But any three dimensional vector can be expressed in an R3, so a three dimensional vector is said to be in R3.) A non-zero vector is a vector that has at least one non-zero entry. As such, [0] is a zero vector while [1] is a non-zero vector. [0] [0] [0] [0] To determine if two vectors are orthogonal (perpendicular), we take the dot product of them (the dot product is the sum of the products of corresponding entries, from 1 to k, of k dimensional vectors). If and only if the dot product of the vectors is equal to 0, the vectors are orthogonal. For example, let's see if the two vectors below are orthogonal. We multiply each corresponding entry, then add up the results to see if we get zero: [2] [3] 2 * 3 = 6 [3] [4] 3 * 4 = 12 [-2] [9] -2 * 9 = -18 (6) + (12) + (-18) = 0, so these two vectors are orthogonal Has this helped you? If you have questions on any of this, or other questions, please write me back. Good luck! - Doctor Jordan, The Math Forum http://mathforum.org/dr.math/ |
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