12 Vectors and the Geometry of Space Copyright Cengage Learning All rights reserved 12 4 The Cross Product Copyright Cengage Learning All rights reserved The Cross Product Learning objectives Find normal vector to a plane through three distinct points Calculate cross product of two three dimensional vectors Calculate area of parallelogram spanned by two vectors and volume of parallelepiped spanned by three vectors Compute scalar triple products and decide whether three given vectors are coplanar 3 How to find a normal vector Given three distinct points in the plane how can we find a normal vector n 4 The Cross Product Notice 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 5 The Cross Product The 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 nonzero vector orthogonal to the plane spanned by a and b 6 The Cross Product In order to make Definition 4 easier to remember we use the notation of determinants A determinant of order 2 is defined by For example 7 The Cross Product A 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 8 The Cross Product Notice also the minus sign in the second term For example 1 0 4 2 6 5 1 12 0 38 9 The Cross Product If 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 is 10 The Cross Product In view of the similarity between Equations 5 and 6 we often write Although 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 11 The Cross Product If 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 1 12 The Cross Product It 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 13 The Cross Product Since 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 14 The Cross Product The geometric interpretation of Theorem 9 can be seen by looking at Figure 2 Figure 2 15 The Cross Product If 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 16 Example Find 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 17 The Cross Product The following theorem summarizes the properties of vector products 18 Triple Products 19 Triple Products The 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 20 Triple Products The 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 Figure 3 The area of the base parallelogram is A b c 21 Triple Products If 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 22 Triple Products If 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 23 Torque 24 Torque The 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 4 25 Torque The torque relative to the origin is defined to be the cross product of the position and force vectors r F and measures the tendency of the body to rotate about the origin The direction of the torque vector indicates the axis of rotation 26
View Full Document
Unlocking...