Unformatted text preview: VECTORS IN Rn
´ ´ ´ JOSE MALAGON-LOPEZ A vector is a physical quantity that is described by both a number (its magnitude) and a direction. We act on such objects by scalars, which are physical quantities that can be described by numbers. Vectors in R2 Recall that any point in the xy -plane, now to be denoted as R2 , represents a vector whose initial point is the origin. If v = (v1, v2) is a vector, then we say that v1 and v2 are the components of v . With this notation, two vectors are equal if they are equal component-wise. We can deﬁne: • Addition: (v1, v2) + (w1, w2) = (v1 + w1, v2 + w2). • Subtraction: (v1, v2) − (w1, w2) = (v1 − w1, v2 − w2 ). • Scalar Multiplication: α(v1, v2) = (α v1 , α v2), where α is a scalar (a real number in this case). Remark. αv is the vector with the same (opposite) direction as v , but α times as long as v if α > 0 (α < 0, respectively). Distinguished Vectors. No all vectors were made equal. We will pay special attention to the following: • The Zero Vector: 0 = (0, 0). • The Standard Unit Vectors: e1 = (1, 0) and e2 = (0, 1). Norm and Dot Product. For any vector v = (v1, v2), its norm is deﬁned as 2 2 v = v1 + v2 . Properties. • v ≥ 0. • α v =| α | v . 1 Remark. We have that ei = 1, for i = 1, 2. Also notice that the deﬁnition of norm is given by Pythagoras. The closer to a “product” of vectors that we have is the following: the Dot Product of v = (v1, v2) and w = (w1, w2) is deﬁned as v • w = v 1 w1 + v 2 w2 . Remark. v • w is always a number, not a vector! Properties. • • • • • v • w = w • v. v • 0 = 0. v • v = v 2. (αv ) • w = α (v • w) = v • (αw). u • (v + w ) = u • v + u • w . Notice that e1 •e2 = 0. This property is a particular case of the following result. Theorem 1. v is perpendicular to w if and only if v • w = 0. Proof. v being perpendicular to w is equivalent to having v−w = v+w . Computing both sides of the equation using the properties above we conclude that this is equivalent to having −2v • w = 2v • w. Which is equivalent to the condition v • w = 0. QED. Projection Along a Vector. Let v and w be two vectors, w = 0. Goal: Write v as the sum of two vectors v = P + Q, with P parallel to w, and Q perpendicular to w. Notice that: (1) v = P + Q implies Q = v − P . (2) Condition “P parallel to w” implies that there should be a scalar α such that P = αw. (3) Condition “Q perpendicular to w” implies that Q • w = 0.
2 All these considerations imply that 0 = Q • w = v − P • w = (v − α w ) • w = (v • w ) − α (w • w ) . Thus, α =
v •w w •w , and P= v•w w. w•w This vector is called the projection of v along w and denoted as projw (v ). The scalar α obtained above is called the component of v along w. Geometric Interpretation of Dot Product. Assume v = 0. Any two vectors v and w deﬁne a unique angle θ, with 0 ≤ θ ≤ π , called the angle between v and w.
θ w Proj W (v) We know that cos(θ) = projw (v) = v v•w w•w w v•w = . v vw As a consequence we have that: • If v • w ≤ 0 then π/2 ≤ θ ≤ π . • If v • w ≥ 0 then π/2 ≥ θ ≥ 0. General Case, Rn There is no need to stop in R2 . Let n ≥ 1. The n-th space Rn is the collection of all the ordered n-truples of the form (v1, v2, . . . , vn), where each vi is a real number.
3 We will denote as v1 v2 v=. . . vn the vector deﬁned by the point (v1, v2, . . . , vn). Sometimes we will use the “transpose” notation v = v1 v2 . . . vn
T T .
T Deﬁnition. Let v = v1 . . . vn and w = w1 . . . wn vectors in Rn . Let α be an scalar (a real number). We deﬁne the vectors in Rn : • v ± w = v 1 ± w1 . . . v n ± wn T • αv = αv1 . . . αvn . We deﬁne the real numbers: • Dot Product: v • w = v1 w1 + · · · + vnwn . 2 2 • Norm: v = v1 + · · · + vn . Proceeding as before we also obtain the identity: cos(θ) = v•w , vw
T be two . where θ is the angle between v and w. Old and New Concepts in Rn . (1) A unit vector is a vector of norm 1. (2) Two vectors are orthogonal if their dot product is zero. (3) Two vectors are orthonormal if they are orthogonal and both are unit vectors. (4) The normalization of a vector v is given by v . v (5) Two vectors v and w are collinear if there is a non-zero scalar α such that v = αw. If such α does not exists, we say that v and w are non-collinear.
4 Distinguished Vectors. (1) The Zero Vector: 0 = 0 . . . 0 . (2) The Standard Unit Vectors: e1 = 1 0 . . . 0 T en = 0 0 . . . 1 .
T T , ... , Properties. The basic properties of vectors are given by the following results. Theorem 2. Let u, v and w be vectors in Rn . Let α and β be scalars. Then (1) (2) (3) (4) (5) (6) (7) (8) (9) v + w = w + v. (u + v ) + w = u + (v + w). v + 0 = v. v + ( −v ) = 0 . (αβ )v = α(βv) = β (αv ). (α + β )v = αv + βv . α(v + w) = αv + αw. 1v = v . 0v = 0. Theorem 2. Let u, v and w be vectors in Rn . Let α be a scalar. Then (1) (2) (3) (4) (5) (6) (7) v ≥ 0. αu =| α | v . v 2 = v • v. v • w = w • v. v • 0 = 0. (αv) • w = α(v • w) = v • (αw). u • (v + w ) = u • v + uw . E-mail address : [email protected] 5 ...
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