PowerPoint PresentationSlide 2The Cross ProductHow to find a normal vector?Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16ExampleSlide 18Slide 19Triple ProductsSlide 21Slide 22Slide 23Slide 24TorqueSlide 26Copyright © Cengage Learning. All rights reserved. 12Vectors and the Geometry of SpaceCopyright © Cengage Learning. All rights reserved. 12.4The Cross Product3The Cross ProductLearning objectives:Find normal vector to a plane (through three distinct points)Calculate cross product of two three-dimensional vectorsCalculate area of parallelogram spanned by two vectors, and volume of parallelepiped spanned by three vectorsCompute scalar triple products, and decide whether three given vectors are coplanar4How to find a normal vector?Given three distinct points in the plane, how can we find a normal vector n?5The Cross ProductNotice that the cross product a b of two vectors a and b, unlike the dot product, is a vector. For this reason it is also called the vector product.Keep in mind that a b is defined only when a and b are three-dimensional vectors.Why is a b defined in this way? What does it mean?6The Cross ProductThe above formula for the cross product is obtained as a solution to the system of linear equations a c = 0 and b c = 0. (You can check this in Section 12.4 of your book.)Therefore the cross product a b is perpendicular to both a and b. (We will confirm this fact later today.) It is perhaps the most important property of the cross product.In particular, if a and b are not parallel, then a b is a non-zero vector orthogonal to the plane spanned by a and b.7The Cross ProductIn order to make Definition 4 easier to remember, we use the notation of determinants.A determinant of order 2 is defined byFor example,8The Cross ProductA determinant of order 3 can be defined in terms of second-order determinants as follows:Observe that each term on the right side of Equation 5 involves a number ai in the first row of the determinant, and ai is multiplied by the second-order determinant obtained from the left side by deleting the row and column in which ai appears.9The Cross ProductNotice also the minus sign in the second term. For example, = 1(0 – 4) – 2(6 + 5) + (–1)(12 – 0) = –3810The Cross ProductIf we now rewrite Definition 4 using second-order determinants and the standard basis vectors i, j, and k, we see that the cross product of the vectors a = a1 i + a2 j + a3 k and b = b1 i + b2 j + b3 k is11The Cross ProductIn view of the similarity between Equations 5 and 6, we often writeAlthough the first row of the symbolic determinant in Equation 7 consists of vectors, if we expand it as if it were an ordinary determinant using the rule in Equation 5, we obtain Equation 6. The symbolic formula in Equation 7 is probably the easiest way of remembering and computing cross products.12The Cross ProductIf a and b are represented by directed line segments with the same initial point (as in Figure 1), then the cross product a b points in a direction perpendicular to the plane through a and b.The right-hand rule gives the direction of a b.Figure 113The Cross ProductIt turns out that the direction of a b is given by the right-hand rule: If the fingers of your right hand curl in the direction of a rotation (through an angle less than 180) from to a to b, then your thumb points in the direction of a b.Now that we know the direction of the vector a b, the remaining thing we need to complete its geometric description is its length | a b |. This is given by the following theorem.14The Cross ProductSince a vector is completely determined by its magnitude and direction, we can now say that a b is the vector that is perpendicular to both a and b, whose orientation is determined by the right-hand rule, and whose length is | a | | b |sin . In fact, that is exactly how physicists define a b.15The Cross ProductThe geometric interpretation of Theorem 9 can be seen by looking at Figure 2.Figure 216The Cross ProductIf a and b are represented by directed line segments with the same initial point, then they determine a parallelogram with base | a |, altitude | b |sin , and area A = | a |(| b |sin ) = | a b |Thus we have the following way of interpreting the magnitude of a cross product.17ExampleFind the area of the triangle with vertices P (1, 4, 6), Q (–2, 5, –1), and R (1, –1, 1).Solution:You can check that PQ PR = –40, –15, 15. The area of the parallelogram with adjacent sides PQ and PR is the length of this cross product:The area A of the triangle PQR is half the area of this parallelogram, that is, .18The Cross ProductThe following theorem summarizes the properties of vector products.19 Triple Products20Triple ProductsThe product a (b c) that occurs in Property 5 is called the scalar triple product of the vectors a, b, and c. Notice that we can write the scalar triple product as a determinant:21Triple ProductsThe geometric significance of the scalar triple product can be seen by considering the parallelepiped determined by the vectors a, b, and c. (See Figure 3.)The area of the base parallelogram is A = | b c |.Figure 322Triple ProductsIf is the angle between a and b c, then the height h of the parallelepiped is h = | a || cos |. (We must use | cos | instead of cos in case > /2.) Therefore the volume of the parallelepiped is V = Ah = | b c || a || cos | = | a (b c) |Thus we have proved the following formula.23Triple ProductsIf we use the formula in and discover that the volume of the parallelepiped determined by a, b, and c is 0, then the vectors must lie in the same plane; that is, they are coplanar.Conversely, if the three vectors are coplanar, then their scalar triple product a (b c) is 0.In particular, a (a b) = 0 and b (a b) is 0, that is, the cross product a b is perpendicular to both a and b.24 Torque25TorqueThe idea of a cross product occurs often in physics. In particular, we consider a force F acting on a rigid body at a point given by a position vector r. (For instance, if we tighten a bolt by applying a force to a wrench as in Figure 4, we produce a turning effect.)Figure 426TorqueThe torque (relative to the origin) is defined to be the cross product of the position and force vectors = r Fand measures the
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