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Extra resources for Elementary Linear Algebra
Subtract p from all these points to obtain the directed line segment consisting of the points 0 + t (q − p) , t ∈ [0, 1] . The point in Rn , q − p, will represent the vector. com 39 Elementary Linear Algebra Fn origin, 0. For example, see the following picture. ✍ ✍ ✍ In this way vectors can be identiﬁed with points of Rn . 2 Let x = (x1 , · · · , xn ) ∈ Rn . The position vector of this point is the vector whose point is at x and whose tail is at the origin, (0, · · · , 0). If x = (x1 , · · · , xn ) is called a vector, the vector which is meant is this position vector just described.
The tension in the cable is the force exerted on this point by the part of the cable nearer the car as well as the force exerted on this point by the part of the cable nearer the tree. 1 The Dot Product There are two ways of multiplying vectors which are of great importance in applications. The ﬁrst of these is called the dot product, also called the scalar product and sometimes the inner product. 1 Let a, b be two vectors in Rn deﬁne a · b as a·b≡ n ∑ a k bk . k=1 The dot product a · b is sometimes denoted as (a, b) where a comma replaces ·.
Vectors are used to model force and other physical vectors like velocity. What was just described would be called a force vector. It has two essential ingredients, its magnitude and its direction. Geometrically think of vectors as directed line segments or arrows as shown in the following picture in which all the directed line segments are considered to be the same vector because they have the same direction, the direction in which the arrows point, and the same magnitude (length). ✍ ✍ ✍ ✍ Because of this fact that only direction and magnitude are important, it is always possible to put → be a directed line segment or vector.