Message Board Assignment (Ch8) 9) The Cross Product is a way to take two three-dimensional vectors and create a new vector which is normal (perpendicular/ orthogonal) to both of the original vectors. It is denoted by u X v. To find the cross product of two vectors u = (U1, U2, U3) and v = (v1, V2, V3), you must take the determinant of a 3x3 matrix which has the standard basis (1, j, k)as its first row and u and v as its second and third rows. As an example, the Cross Product of u = (2, -1,0) and v = (3,4,5) is given by the following: uxv = 14 $12+ 13 3/1 + 13 7/8 = - 52-103 +12k 4 5 So ux v = (-5, -10, 11), and this vector is normal to both u and v (you can verify this using the dot product). Use the Cross Product to complete the following problems: a) Find a unit vector which is normal to both u = (7,2,5) and v = (-1, -3,6). b) Recall that a normal vector can be used to find the equation of a plane. Find the equation of a plane which is parallel to both u = (1,2, -3) and v = (-4,5, -6) and goes through the point (-2,8,7). c) Explain in words how we know that u . (u x v) = v . (u x v) for any vectors u and v. d) Suppose that we have a sphere which is tangent to a plane and both the sphere and the plane meet at the origin. If the points (4,7, -2) and (3, -2,5) are points on the plane and the radius of the sphere is 6, find the equation of the sphere. [Note: There are two possible answers depending on which direction you choose for your normal vector. Either is fine.]