AAE 203: Aeromechanics I Vectors (part 2) Dengfeng Sun dsun@purdue.edu August 30, 2010 Reference: AAE 203 Notes, by Prof. Martin Corless. 1/9 Components Suppose ¯b1,¯b2, are any pair of non-zero, non-parallel vectors in a plane. Then, for every vector ¯V in the plane, there is a unique pair of scalars, V1, V2 such that ¯V = V1 ¯b1 + V2 ¯b2 The pair (¯b1,¯b2) of vectors is called a basis. It defines a coordinate system. With respect to this basis, V1¯b1 and V2¯b2 are called the vector components of ¯V . The scalars V1 and V2 are called the scalar components or coordinates of ¯V . A basis permits one to represent uniquely any vector ¯V in the plane as a pair of scalars (V1, V2). 2/9 Examples of basis b2 2V V 1V b1 e^2 e^1 P AO V2 V1 V ? 3/9 Perpendicular components ¯V = V1?e1 + V2?e2 V1 = V cos?, V2 = V sin? V = q V 21 + V 22 tan? = V2/V1 e^2 e^1 V ? ? V e^2 e^1 4/9 Some definitions Suppose ¯b1,¯b2 and ¯b3 are any three non-zero vectors which are not parallel to a common plane. Then, given any vector ¯V , there exists a unique triplet of scalars, V1,V2,V3 such that ¯V = V1¯b1 + V2¯b2 + V3¯b3 . The triplet of vectors, (¯b1, ¯b2, ¯b3), is called a basis. It defines a coordinate system. With respect to this basis, the vectors V1¯b1,V2¯b2,V3¯b3 are the vector components of ¯V and the scalars V1,V2,V3 are the scalar components or coordinates of ¯V . The most important thing about a basis is that it permits one to represent uniquely any vector ¯V as a triplet of scalars (V1,V2,V3). In this course, we consider mainly a special case, namely the case in which ¯b1,¯b2,¯b3 are mutually perpendicular unit vectors. 5/9 Mutually ? components Let ?e1,?e2,?e3 be any three mutually orthogonal (perpendicular) unit vectors. We call (?e1,?e2,?e3) an orthogonal triad. Since (?e1,?e2,?e3) constitute a basis, any vector ¯V can be uniquely resolved into components parallel to ?e1, ?e2, ?e3, that is, there are unique scalars V1,V2,V3 such that ¯V = V1?e1 + V2?e2 + V3?e3 6/9 Mutually ? components The vectors V1?e1,V2?e2,V3?e3 are called rectangular components. The scalars V1,V2,V3 are called rectangular scalar components or rectangular coordinates. V = radicalBig V 21 + V 22 + V 23 where V = |¯V|. 7/9 + and × of scalar components Addition of vectors via addition of scalar components ¯V = V1?e1 + V2?e2 + V3?e3 ¯W = W1?e1 + W2?e2 + W3?e3 ¯V + ¯W = (V1+W1)?e1+(V2+W2)?e2+(V3+W3)?e3 Scalar multiplication of a vector via multiplication of its scalar components ¯V = V1?e1 + V2?e2 + V3?e3 k¯V = (kV1)?e1 + (kV2)?e2 + (kV3)?e3 8/9 Example Given ¯V and the orthogonal triad (¯u1, ?u2, ?u3) as shown, find (i) scalars V1, V2, V3, such that ¯V = V1?u1 + V2?u2 + V3?u3 (ii) scalars n1, n2, n3, such that ?u¯V = n1?u1 + n2?u2 + n3?u3 . 9/9
