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## Chapter 12 – Vectors and the Geometry of Space

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**Chapter 12 – Vectors and the Geometry of Space**12.4 The Cross Product 12.4 The Cross Product**Definition – Cross Product**Note: The result is a vector. Sometimes the cross product is called a vector product. This only works for three dimensional vectors. 12.4 The Cross Product**Cross Product as Determinants**• To make Definition one easier, we will use the notation of determinants. A determinant of order 2 is defined by 12.4 The Cross Product**Cross Product as Determinants**• A determinant of order 3 is defined in terms of second order determinates as shown below. 12.4 The Cross Product**Cross Product as Determinants**12.4 The Cross Product**Example 1 – pg. 814 #5**• Find the cross product a x b and verify that it is orthogonal to both a and b. 12.4 The Cross Product**Theorem 5**• The direction of axb is given by the right hand rule: If your fingers of your right hand curl in the direction of a rotation of an angle less than 180o from a to b, then your thumb points in the direction of axb. 12.4 The Cross Product**Visualization**• The Cross Product 12.4 The Cross Product**Theorems**12.4 The Cross Product**Example 2**• For the below problem, find the following: • a nonzero vector orthogonal to the plane through the points P, Q, and R. • the area of triangle PQR. P(2,1,5) Q(-1,3,4) R(3,0,6) 12.4 The Cross Product**Theorem 8**• Note: The cross product is not commutative i x j j x i • Associative law for multiplication does not hold. (a x b) x c a x (b x c) 12.4 The Cross Product**Definition – Triple Products**• The product a (b x c) is called the scalar triple product of vectors a, b, and c. We can write the scalar triple product as a determinant: 12.4 The Cross Product**Definition – Volume of a Parallelepiped**12.4 The Cross Product**Example 3 – pg. 815 # 36**• Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. P(3,0,1) Q(-1,2,5) R(5,1,-1) S(0,4,2) 12.4 The Cross Product**Torque**• Cross product occurs often in physics. • Let’s consider a force, F, acting on a rigid body at a point given by a position vector r. (i.e. tightening a bolt by applying force to a wrench). The torque is defined as = r x F and measures the tendency of the body to rotate about the origin. • The magnitude of the torque vector is ||= |r x F| = |r||F|sin 12.4 The Cross Product**Example 4 – pg. 815 #41**• A wrench 30 cm long lies along the positive y-axis and grips a bolt at the origin. A force is applied in the direction <0,3,-4> at the end of the wrench. Find the magnitude of the force needed to supply 100 Nm of torque to the bolt. 12.4 The Cross Product**More Examples**The video examples below are from section 12.4 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. • Example 1 • Example 2 • Example 5 12.4 The Cross Product**Demonstrations**Feel free to explore these demonstrations below. • Cross Product of Vectors 12.4 The Cross Product