-[ (1,2) -2 (-1,-2) FIGURE 1 (1,-3) Vectors Are Directed Line Segments The Scalar Product of Two Vectors An important result of multiplying two vectors is the scalar product.To define the scalar prod- uct of two vectors,suppose we have a row vector u=[u u2..u]and a column vector 2 : of the same dimension.The scalar product of u and v(written u.v)is the number 4M+22+…+山nym For the scalar product of two vectors to be defined,the first vector must be a row vec- tor and the second vector must be a column vector.For example,if u=[123] and V= 21 2 then u.v=1(2)+2(1)+3(2)=10.By these rules for computing a scalar product,if u- and v=[23] then u.v is not defined.Also,if u=[123] and then u.v is not defined because the vectors are of two different dimensions. Note that two vectors are perpendicular if and only if their scalar product equals 0. Thus,the vectors [1 -1]and [1 1]are perpendicular. We note that u v=uv cos 0,wherelul is the length of the vector u and 0 is the angle between the vectors u and v.The Scalar Product of Two Vectors An important result of multiplying two vectors is the scalar product. To define the scalar prod￾uct of two vectors, suppose we have a row vector u = [u1 u2  un] and a column vector v of the same dimension. The scalar product of u and v (written u  v) is the number u1v1  u2v2    unvn. For the scalar product of two vectors to be defined, the first vector must be a row vec￾tor and the second vector must be a column vector. For example, if u [1 2 3] and v then u  v 1(2)  2(1)  3(2) 10. By these rules for computing a scalar product, if u and v [2 3] then u  v is not defined. Also, if u [1 2 3] and v then u  v is not defined because the vectors are of two different dimensions. Note that two vectors are perpendicular if and only if their scalar product equals 0. Thus, the vectors [1 1] and [1 1] are perpendicular. We note that u  v u v cos u, where u is the length of the vector u and u is the angle between the vectors u and v. 3 4 1 2 2 1 2 v1 v2    vn 2.1 Matrices and Vectors 13 3 2 x2 x1 1 – 1 – 2 – 2 w u v – 1 (–1, –2) (1, 2) u = 1 2 v = w = (1, –3) 1 2 – 3 1 –3 –2 –1 FIGURE 1 Vectors Are Directed Line Segments