Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product Chapter 12: Vectors and the Geometry of Space Math 213 Cross Product Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product Let u = hu1 , u2 , u3 i and v = hv1 , v2 , v3 i. Then the cross product of u and v is u ⇥ v = hu2 v3 u3 v2 , u3 v1 u1 v3 , u1 v2 u2 v1 i Cross Product Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product Let u = hu1 , u2 , u3 i and v = hv1 , v2 , v3 i. Then the cross product of u and v is u ⇥ v = hu2 v3 u3 v2 , u3 v1 u1 v3 , u1 v2 u2 v1 i Important Notes: The cross product of two vectors is a vector. The cross product is only defined for vectors in R3 . Cross Product Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product Let u = hu1 , u2 , u3 i and v = hv1 , v2 , v3 i. Then the cross product of u and v is u ⇥ v = hu2 v3 u3 v2 , u3 v1 u1 v3 , u1 v2 u2 v1 i Important Notes: The cross product of two vectors is a vector. The cross product is only defined for vectors in R3 . We remember this formula by writing u⇥v= i j k u1 u2 u3 v1 v2 v3 Even though this doesn’t really make sense, it’s a tool to help us remember. Example The cross product of two vectors Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product Let u = h2, 3, 1i, v = h3, 1, 4i. Compute u ⇥ v, v ⇥ u, and u ⇥ u. Example The cross product of two vectors Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product Let u = h2, 3, 1i, v = h3, 1, 4i. Compute u ⇥ v, v ⇥ u, and u ⇥ u. h2, 3, 1i ⇥ h3, 1, 4i = i 2 3 = h 12 j k 3 1 1 4 1, (8 = h 13, 5, 11i 3), 2 ( 9)i Example The cross product of two vectors Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product Let u = h2, 3, 1i, v = h3, 1, 4i. Compute u ⇥ v, v ⇥ u, and u ⇥ u. h2, 3, 1i ⇥ h3, 1, 4i = i 2 3 j k 3 1 1 4 = h 12 1, (8 3), 2 ( 9)i = h 13, 5, 11i h3, 1, 4i ⇥ h2, 3, 1i = i 3 2 = h1 j k 1 4 3 1 ( 12), (3 = h13, 5, 11i 8), 9 2i Example Solution (continued) The cross product of two vectors Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product So we see that u ⇥ v 6= v ⇥ u. In fact, in our example (we will see later this holds in general), u ⇥ v = (v ⇥ u). Example Solution (continued) The cross product of two vectors Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product So we see that u ⇥ v 6= v ⇥ u. In fact, in our example (we will see later this holds in general), u ⇥ v = (v ⇥ u). h2, 3, 1i ⇥ h2, 3, 1i = i 2 2 =h 3 j k 3 1 3 1 ( 3), (2 = h0, 0, 0i 2), 6 ( 6)i Example Solution (continued) The cross product of two vectors Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product So we see that u ⇥ v 6= v ⇥ u. In fact, in our example (we will see later this holds in general), u ⇥ v = (v ⇥ u). h2, 3, 1i ⇥ h2, 3, 1i = i 2 2 =h 3 j k 3 1 3 1 ( 3), (2 2), 6 ( 6)i = h0, 0, 0i It is also true in general that u ⇥ u = 0 for any vector u 2 R3 . Orthogonality Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product Theorem The vector u ⇥ v is orthogonal to both u and v. Orthogonality Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product Theorem The vector u ⇥ v is orthogonal to both u and v. Proof. To see u ⇥ v is orthogonal to u, we compute the dot product of u ⇥ v and u: u ⇥ v · u = (u2 v3 = u1 u2 v3 u3 v2 )u1 + (u3 v1 u1 u3 v2 + u2 u3 v1 u1 v3 )u2 + (u1 v2 u1 u2 v3 + u1 u3 v2 u2 v1 )u3 u2 u3 v1 =0 A similar calculation shows that u ⇥ v is also orthogonal to v. Right-hand Rule Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product If a and b are positioned such that their tails have the same initial point, then the direction that the cross product points is given by a right-hand rule. If you curl the fingers of your right hand from a to b (through an angle less than 180 ), then your thumb points in the direction of a ⇥ b. Magnitude of Cross Product Vector Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product Theorem If ✓ is the angle between a and b (so 0 ✓ ⇡), then |a ⇥ b| = |a||b| sin ✓ (Proof is in text but an elaborated version is found in “Area of parallelogram proof” on Canvas). Magnitude of Cross Product Vector Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product Theorem If ✓ is the angle between a and b (so 0 ✓ ⇡), then |a ⇥ b| = |a||b| sin ✓ (Proof is in text but an elaborated version is found in “Area of parallelogram proof” on Canvas). The area of the parallelogram above is A = |a||b| sin ✓. Therefore, the length of the cross product vector a ⇥ b is equal to the area of the parallelogram determined by a and b. Application of Previous Theorem Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product Another application of this theorem is to use it to find the distance between a line ` and a point P not on the line. We pick two points Q and R on the line and find the vectors ! ! QR and QP. Application of Previous Theorem Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product Another application of this theorem is to use it to find the distance between a line ` and a point P not on the line. We pick two points Q and R on the line and find the vectors ! ! QR and QP. The distance D between P and ` is ! D = |QP| sin ✓ ! ! where ✓ is the angle between QP and QR. But ! ! ! ! |QP ⇥ QR| = |QP||QR| sin ✓ Therefore ! ! |QP ⇥ QR| D= ! |QR| Properties of the Cross Product Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product Let u, v, w 2 R3 , and let k 2 R. Then 1 2 3 4 5 6 u⇥v= v⇥u (k u) ⇥ v = k (u ⇥ v) = u ⇥ (k v) u ⇥ (v + w) = u ⇥ v + u ⇥ w (u + v) ⇥ w = u ⇥ w + v ⇥ w u · (v ⇥ w) = (u ⇥ v) · w u ⇥ (v ⇥ w) = (u · w)v (distributive from the left) (distributive from the right) (scalar triple product) (u · v)w Properties of the Cross Product Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product Let u, v, w 2 R3 , and let k 2 R. Then 1 2 3 4 5 6 u⇥v= v⇥u (k u) ⇥ v = k (u ⇥ v) = u ⇥ (k v) u ⇥ (v + w) = u ⇥ v + u ⇥ w (u + v) ⇥ w = u ⇥ w + v ⇥ w u · (v ⇥ w) = (u ⇥ v) · w u ⇥ (v ⇥ w) = (u · w)v (distributive from the left) (distributive from the right) (scalar triple product) (u · v)w The proofs of all of these properties follow from the definitions of the cross product and dot product. Scalar Triple Product Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product We can write the scalar triple product, a · (b ⇥ c) as the following determinant: a · (b ⇥ c) = a1 a2 a3 b1 b2 b3 c1 c2 c3 Scalar Triple Product Chapter 12: Vectors and the Geometry of Space Consider the parallelepiped determined by the vectors a, b, and c as pictured below. Math 213 Section 4: The Cross Product The area of the base parallelogram is A = |b ⇥ c|. Scalar Triple Product Chapter 12: Vectors and the Geometry of Space Consider the parallelepiped determined by the vectors a, b, and c as pictured below. Math 213 Section 4: The Cross Product The area of the base parallelogram is A = |b ⇥ c|. The height h of the parallelepiped is |a||cos ✓| where ✓ is the angle between b ⇥ c and a. (We have to use |cos ✓| in case ✓ > ⇡2 .) Scalar Triple Product Chapter 12: Vectors and the Geometry of Space Consider the parallelepiped determined by the vectors a, b, and c as pictured below. Math 213 Section 4: The Cross Product The area of the base parallelogram is A = |b ⇥ c|. The height h of the parallelepiped is |a||cos ✓| where ✓ is the angle between b ⇥ c and a. (We have to use |cos ✓| in case ✓ > ⇡2 .) So the volume of the parallelepiped is V = Ah = |b ⇥ c||a||cos ✓| = |a · (b ⇥ c)| Homework Section 12.4 Chapter 12: Vectors and the Geometry of Space Math 213 Section 4: The Cross Product 1-37 eoo, 53