2313-1-5 - S ECTION 1.5 1.5 T HE C ROSS P RODUCT 17 T HE C...

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S ECTION 1.5 T HE C ROSS P RODUCT 17 1.5. T HE C ROSS P RODUCT As we saw in the last section, it is easiest to write down an equation for a plane if we know its normal vector. In order to find the equation of a plane containing two given non-skew lines, then, we want to be able to find a vector orthogonal to both lines. In this section we introduce a new operation on vectors, called the cross product , and written a b , which does this (and more). Instead of the usual approach of presenting the somewhat mysterious definition of the cross product and then studying its properties, we are going to go in the other direction. We are going to write down our goals for this operation, and then use these to figure out the formula. Since all of these goals are going to turn out to hold for the cross product, we are going to call them properties, but the reader should keep in mind that they are just desires until we prove them. Unlike the dot product which produces a scalar, the cross product will produce a vector . We will create the cross product so that this new vector, a b , is orthogonal to both a and b , but this raises a question: which direction? For example, should i j be k or k ? Let’s decide that our goal will be to have the cross product agree with the right-hand rule (or book rule). This means that if you point you right arm in the direction of a and then curl your fingers in towards your palm in the direction of b (at an angle less than 180 ), then your thumb points in the direction of a b . Orthogonality Property of Cross Products. The cross product a b is orthogonal to both a and b , and points in the direction given by the right-hand rule. Because we are insisting on this property, note that the cross product will not be com- mutative. Going back to i and j : i j k , but j i k . In fact, the cross product will turn out to always be anti-commutative : Anti-Commutativity Property of Cross Products. For all vectors a and b , a b b a . It is good practice to try all the cross products of basis vectors at this point. One will find the following, by using the right-hand rule: i j k , i k j , j i k , j k i , k i j , k j i ,
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18 C HAPTER 1 V ECTORS AND V ECTOR -V ALUED F UNCTIONS But what about the diagonal entries, like i i ? If we follow the Anti-Commutativity Prop- erty of cross products, then we have to have i i i i , so we are forced to set i i 0 (the zero vector). More generally, let us make the following goal.
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