Calculus 3 Section 12.4 Reading Assignment: The Cross Product

Answer Only Exercise 1 by using a screenshot provided Calculus Pearson textbook. Make sure you read these three questions very carefully and see on what it is asking for and what is really about. Please be very serious careful with this assignment of exercise #1.

References: Thomas' Calculus: Early Transcendentals | Calculus | Calculus | Mathematics | Store | Pearson+

Exercise 1. Read the subsections "The Cross Product of Two Vectors in Space" (p. 736 - 737) and "|u v| is the Area of a Parallelogram" (p. 737 - 738). Explain the geometric meaning of the cross product. For convenience, assume that the vectors are not parallel. Be thorough in your explanation and explain the meaning of the word "normal" if you use it in your answer. Both the magnitude and direction should be described in your answer.

The cross product can be used to find the angle between two vectors, just like the dot product, but it's easier (and better) to use the dot product for that.

The determinant formula used here is useful for remembering how to compute the cross product. Using the 2x2 and 3x3 determinant formulas do come up often, so they're worth refreshing yourself on them (or learning if this is your first encounter with them).

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12.4 The Cross Product In studying lines in the plane, when we needed to describe how a line was tilting, we used U X the notions of slope and angle of inclination. In space, we want a way to describe how a plane is tilting. We accomplish this by multiplying two vectors in the plane together to get a third vector perpendicular to the plane. The direction of this third vector tells us the "inclination" of the plane. The product we use to multiply the vectors together is the vector or cross product, the second of the two vector multiplication methods. The cross product gives us a simple way to find a variety of geometric quantities, including volumes, areas, and perpendicular vectors. We study the cross product in this section. The Cross Product of Two Vectors in Space FIGURE 12.28 The construction of We start with two nonzero vectors u and v in space. Two vectors are parallel if one is a u X V. nonzero multiple of the other. If u and v are not parallel, they determine a plane. The vec- tors in this plane are linear combinations of u and v. so they can be written as a sum au + by. We select the unit vector n perpendicular to the plane by the right-hand rule. This means that we choose n to be the unit (normal) vector that points the way your right thumb points when your fingers curl through the angle O from u to v (Figure 12.28). Then we define a new vector as follows. DEFINITION The cross product u X v ("u cross v") is the vector uxv = (july sin #) n.Chapter 12 Vectors and Geometry of Space 12.4 The Cross Product 737 Unlike the dot product, the cross product is a vector. For this reason it is also called the vector product of u and v. and can be applied only to vectors in space. The vector u X v is orthogonal to both u and v because it is a scalar multiple of n. There is a straightforward way to calculate the cross product of two vectors from their components. The method does not require that we know the angle between them (as sug- gested by the definition), but we postpone that calculation momentarily so we can focus first on the properties of the cross product. Since the sines of 0 and a are both zero, it makes sense to define the cross product of two parallel nonzero vectors to be 0. If one or both of u and v are zero, we also define u X v to be zero. This way, the cross product of two vectors u and v is zero if and only if u and v are parallel or one or both of them are zero. Parallel Vectors Nonzero vectors u and v are parallel if and only if u X v = 0. The cross product obeys the following laws. Properties of the Cross Product If u, v. and w are any vectors and r, s are scalars, then 1. (ru) X (sv) = (rs)(u X v) 2. ux(v+w)=uxvtuxw 3. v Xu = -(ux v) 4. (v+w)Xu=vxu+wxu 5. 0 Xu=0 6. 1 X (v X w) = (u . w)v - (u *v)wTo visualize Property 3, for example, notice that when the fingers of your right hand curl through the angle # from v to u. your thumb points the opposite way; the unit vector we choose in forming v X u is the negative of the one we choose in forming u X v (Figure 12.29). Property 1 can be verified by applying the definition of cross product to both sides of the equation and comparing the results. Property 2 is proved in Appendix 8. Property 4 follows by multiplying both sides of the equation in Property 2 by -1 and reversing the order of the products using Property 3. Property 5 is a definition. As a rule, cross product multiplication is not associative so (u X v) X w does not generally equal u X (v X w). (See Additional Exercise 17.) FIGURE 12.29 The construction of When we apply the definition and Property 3 to calculate the pairwise cross products vXI. of i, j, and k, we find (Figure 12.30) ixj =-( x )) = k j Xk =-(kxj) =i k-ixj k xi = -(i X k) = j and ixi=jx] =kxk =0. jokxi u x v| Is the Area of a Parallelogram Because n is a unit vector, the magnitude of u X v is i-jxk FIGURE 12.30 The pairwise cross lux vl = lullvl Isinella) = lullw| sine. products of i, j. and k.Chapter 12 Vectors and Geometry of Space 738 Chapter 12 Vectors and the Geometry of Space Area = base - height This is the area of the parallelogram determined by u and v (Figure 12.31), Ju| being the - lu| . |v| |sin el base of the parallelogram and | v| | sind| the height. =uxv ih = [visine] Determinant Formula for u X v Our next objective is to calculate u X v from the components of u and v relative to a Cartesian coordinate system. FIGURE 12.31 The parallelogram Suppose that determined by u and v. u =mitmituk and v=withjtuk. Then the distributive laws and the rules for multiplying i, j, and k tell us that oxv= (uitujtuk) X (uituj+uk) =mixi+mixj+mixk +mujxi+muj xj+ umj x k +ukxitnukxjtluk X k = (1213 - muz)i - (un; - uzu )j + (un - uzu)k. The component terms in the last line are hard to remember, but they are the same as the terms in the expansion of the symbolic determinant1 k Determinants 2 x 2 and 3 X 3 determinants ure evaluated as follows: So we restate the calculation in this easy-to-remember form. a b = ad - be Calculating the Cross Product as a Determinant If u = wit uj + uk and v = uji + uj + uk, then C2 i k I X V= - 02 C3 R( - 1, 1, 2) EXAMPLE 1 Find u X v and v X u if u = 21 + j + k and v = -41 + 3j + k. Solution We expand the symbolic determinant: P(1. - 1, 0) -4 3 = -21 - 6j + 10k 0(2. 1, -1) v Xu = -(u * v) = 21 + 6j - 10k Property 3 FIGURE 12.32 The vector PQ X PR is perpendicular to the plane of triangle POR (Example 2). The area of triangle POR is EXAMPLE 2 Find a vector perpendicular to the plane of P(1, -1, 0), 0(2, 1, -1). half of [ PQ x PR| (Example 3). and R(-1, 1, 2) (Figure 12.32)