MAT223H1 Lecture Notes - Lecture 6: Parallelogram, Natural Number, Dot Product
Document Summary
Mat233 lecture 6 unit 2 geometry of vectors. A vector is a matrix with only 1 column. Consider some matrix with 1 column and n rows: (note: to denote vectors, some people put an arrow over the vector but this is usually not necessary: the numbers in the vector are real numbers. {[(cid:1874)(cid:3041)(cid:1874)(cid:3040)]= (cid:1874)(cid:3041), ,(cid:1874)(cid:3040) }= (cid:3040: the e(cid:454)po(cid:374)e(cid:374)t (cid:862)(cid:373)(cid:863) is just a (cid:374)otatio(cid:374) (cid:373)ea(cid:374)i(cid:374)g the set of (cid:448)ectors (cid:449)ith m entries, n is a natural number (integer) [2] we can scale them with constants i. e. multiply a vector v by a real number. These 2 properties work for any vectors with n entries (cid:884) (cid:1874)=(cid:883)(cid:884)[(cid:883)(cid:884)]=[(cid:883) (cid:884)(cid:883) ] Consider the vectors: (cid:1874)=[(cid:1874)(cid:3041)(cid:1874)(cid:3040)] (cid:3040) , (cid:1873)=[(cid:1873)(cid:3041)(cid:1873)(cid:3040)] (cid:3040) (cid:1874)+(cid:1873)=[(cid:1874)(cid:3041)(cid:1874)(cid:3040)]+ [(cid:1873)(cid:3041)(cid:1873)(cid:3040)]= [(cid:1874)(cid:3041) + (cid:1873)(cid:3041) (cid:1874)(cid:3040) + (cid:1873)(cid:3040)] column. Definition 2. 3 dot products & lengths of vectors. [2] we can scale them with constants: here alpha is the notation for a real number. If we compute the length of this vector we get:
