The Cross Product. Philippe B. Laval. Spring 2012 KSU. Philippe B. Laval (KSU) The Cross Product Spring /

Transcription

1 The Cross Product Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) The Cross Product Spring / 15

2 Introduction The cross product is the second multiplication operation between vectors we will study It is very different from the scalar product not only in the way it is computed but also in the result we get and in its applications Philippe B Laval (KSU) The Cross Product Spring / 15

3 Definition Unlike the dot product, the cross product is only defined for 3-D vectors So, in this section, when we use the word vector, we will mean 3-D vector The result is a vector, perpendicular to the two vectors we are taking the cross product of Definition The cross product also called vector product of two vectors u = u x, u y, u z and v = v x, v y, v z, denoted u v, is defined to be u x u y u z v x v y v z = u y v z u z v y u z v x u x v z u x v y u y v x Thus, the cross product of two 3-D vectors is also a 3-D vector This formula is not easy to remember However, if you know about matrices and the determinant of a matrix, the cross product can be expressed in term of them Let us first quickly review what they are Philippe B Laval (KSU) The Cross Product Spring / 15

4 Definitions: Determinant of a Matrix Definition Determinant [ ] of a 2 2 matrix The determinant of a 2 2 matrix a b, denoted by c d a b c d is defined to be Example Find a c b d = ad bc Philippe B Laval (KSU) The Cross Product Spring / 15

5 Definitions: Determinant of a Matrix Definition Determinant of a 3 3 matrix The determinant of a 3 3 matrix a 1 a 2 a 3 a 1 a 2 a 3 b 1 b 2 b 3 denoted by b 1 b 2 b 3 is defined to be c 1 c 2 c 3 c 1 c 2 c 3 a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 = a 1 b 2 b 3 c 2 c 3 a 2 b 1 b 3 c 1 c 3 + a 3 b 1 b 2 c 1 c 2 Example Find = a 1 (b 2 c 3 c 2 b 3 ) a 2 (b 1 c 3 c 1 b 3 ) + a 3 (b 1 c 2 c 1 b 2 ) Philippe B Laval (KSU) The Cross Product Spring / 15

6 Cross Product as a Determinant Lemma If u = u x, u y, u z and v = v x, v y, v z then i j k u v = u x u y u z v x v y v z Which makes it much easier to remember Example For u = 3, 1, 1 and v = 4, 7, 2, compute u v Philippe B Laval (KSU) The Cross Product Spring / 15

7 Properties Theorem Let u and v denote two non-zero vectors Then, the following is true: 1 u u = 0 2 u v is perpendicular to both u and v 3 u v = u v sin α where α is the smallest angle between u and v (0 α π) 4 u v = u v sin α n where n is the unit vector perpendicular to both u and v whose direction is determined by the right-hand rule The above properties tell us that u v is the vector perpendicular to both u and v which direction is given by the right-hand rule and whose magnitude is u v sin α From property 3 of the theorem, it follows that two non-zero vectors are parallel if and only if their cross product is 0 Philippe B Laval (KSU) The Cross Product Spring / 15

8 Examples Example Find a unit vector perpendicular to both u = 1, 1, 1 and v = 2, 1, 0 Example Find a vector perpendicular to the plane containing the three points P : (1, 1, 2), Q : (2, 1, 1) and R : (2, 1, 0) Philippe B Laval (KSU) The Cross Product Spring / 15

9 More Properties The cross product satisfies more properties which we will not prove because they are very tedious Theorem Let u, v, and w be three vectors and a be a scalar The following is true: 1 u v = v u (this tells us that the cross product is not commutative 2 (a u) v = a ( u v) = u (a v) 3 u ( v + w) = u v + u w 4 ( u + v) w = u w + v w Philippe B Laval (KSU) The Cross Product Spring / 15

10 Area of a Parallelogram Consider a parallelogram whose sides are given by the vectors u and v as shown in the figure below Remembering that the area of a parallelogram is the length of its base times its height, we see that the area A of this parallelogram is A = u v sin θ = u v Philippe B Laval (KSU) The Cross Product Spring / 15

11 Examples Example Ẹxpress the area of a triangle as a cross product Example Find the area of the parallelogram shown in the figure below Philippe B Laval (KSU) The Cross Product Spring / 15

12 Triple Products Definition Given three non-zero vectors u, v, and w, the product u ( v w) is called the scalar triple product of the vectors u, v, and w Lemma The volume of the parallelepiped determined by the vectors u, v, and w as shown in the figure below is the magnitude of their scalar triple product that is: u ( v w) Philippe B Laval (KSU) The Cross Product Spring / 15

13 Triple Products Corollary Three non-zero vectors u, v, and w are coplanar (on the same plane) if u ( v w) = 0 Lemma The triple product satisfies: u ( v w) = w ( u v) Lemma The scalar triple product of three non-zero vectors u, v, and w can be computed by calculating the determinant u x u y u z u ( v w) = v x v y v z w x w y w z Philippe B Laval (KSU) The Cross Product Spring / 15

14 Summary The cross product is a very important quantity in mathematics It can be used for: 1 Find a vector perpendicular to two non-zero vectors (often used in computer graphics) 2 Find the area of a parallelogram 3 Find the volume of a parallelepiped 4 Determine if two non-zero vectors are parallel 5 Determine if three non-zero vectors are coplanar 6 Many applications in physics which we will not discuss here Philippe B Laval (KSU) The Cross Product Spring / 15

15 Exercises See the problems at the end of section 14 in my notes on the cross product Philippe B Laval (KSU) The Cross Product Spring / 15