12.1 vectors in the plane We occasionally denote an (x,y) plane by R^2. This notation represents the idea that the plane is a "product” of two copies of the real line R, where one copy represents the points x-coordinates and the other represents the y-coordinates A two-dimensional vector V is determined by two points in the plane: an initial point P (also called the tail or basepoint) and a terminal point Q (also called the head or tip) Vector terminology Two vectors V and W of nonzero length are called parallel if the lines through V and W are parallel. Parallel vectors point either in the same or in opposite directions A vector V is said to undergo a translation when it is moved to begin at a new point without changing its length or direction. The resulting vector W is called a translation of V. A translation W of a vector V has the same length and directions as V but a different basepoint The pair of components determine the length and direction of V, but not its basepoint. Therefore, two vectors are equivalent if and only if they have the same components Definition: Components of a vector • The components of v = PQ, where P = (a1,b1) and Q = (a2,b2), are the quantities °¡¡!° p ° ° kvk = °P Q° = a2 + b2 Vector algebra a = a2 - a1 (x-component) b = b2 - b1 (y-component) The vector sum v + w is defined when V and W have the same basepoint: translate W to the equivalent vector W' whose tail coincides with the head of V The term "scalar" is another word for real number, and we often speak of scalar versus vector quantities. Thus, the number 8 is a scalar, while (8,2) is a vector It [scalar] points in the same direction as V if lambda > 0 It [scalar] points in the opposite direction if lambda < 0 A linear combination of vectors V and w is a vector rv + sw where r and s are scalars. If V and w are not parallel, then every vector U in the plane can be expressed as a Vector operations using linear combination u = rv + sw components A vector of length 1 is called a unit vector. Unit vectors are often used to indicate direction, when it is not necessary to specify length 1 ev = ¢v kvk e = hcos µ; sin µi The vectors i and j are called the standard basis vectors. Every vector in the plane is a linear combination of i and j Triangle inequality: kv + wk · jj v jj + jj wjj If V = and w = , then: • v + w = (a+c, b+d) • v - w = (a-c, b-d) • lambda(v) = (lambda(a), lambda(b)) • v+0=0+v=v