MATH 251 – LECTURE 3 JENS FORSGÅRD http://www.math.tamu.edu/~jensf/ This week: 11.3–6 webAssign: 11.3–5, due 2/1 11:55 p.m. Next week: 11.6–7, 12.1–3 webAssign: 11.6, 12.1, and 12.3, opens 2/1 12 a.m. Help Sessions: M W 5.30–8 p.m. in BLOC 161 Office Hours: BLOC 641C M 12:30–2:30 p.m. W 2–3 p.m. or by appointment. Dot product (or Scalar product) We defined a · b = |a| |b| cos(θ). We’ve deduced that (a + b) · c = a · c + b · c and, similarly, c · (a + b) = c · a + c · b. Let a = ha1, a2, a3i and b = hb1, b2, b3i. Compute a · b without first computing θ. Dot product (or Scalar product) Exercise 1. Let a = h1, 0, 2i and b = h1, −2, 0i. Compute the intermediate angel θ of a and b. Cross product (or Vector product) The Vector product a × b is the unique vector such that (1) a × b is orthogonal to both a and b. (That is, a · c = b · c = 0.) (2) |a × b| is equal to the area of the parallelogram spanned by a and b. (That is, |a × b| = |a||b| sin(θ).) (3) a, b, a × b fulfill the right hand rule. Cross product (or Vector product) Exercise 2. h1, 0, 0i × h0, 1, 0i = Exercise 3. h0, 0, 1i × h0, 1, 0i = Cross product (or Vector product) Properties of the Cross product: (1) a × b = −b × a (2) (ka) × b = k(a × b) = a × (kb) (3) a × (b + c) = a × b + a × c (4) (a + b) × c = a × c + b × c. In coordinates: a × b = h a2b3 − a3b2, a3b1 − a1b3, a2 a3 a1 a3 , − = b1 b3 , b2 b3 Exercise 4. h2, 3, 1i × h−1, 1, 1i = a1b2 − a2b1 i a1 a2 b1 b2 Cross product Exercise 5. Find the area of the triangle with vertices A(1, 1, 1), B(0, 0, 0), and C(2, −1, 4). Cross product Exercise 6. Find the volume of the parallelepiped spanned by the vectors h1, 2, 3i, h4, 2, −3i and h0, −2, 0i. Lines A line L is determined by a point P0 = P0(x0, y0, z0) and a direction v. That is, the position vector r of any point P on v can be written as r = r0 + tv where r0 is the position vector of P0 and t is a real parameter. This equations is called a parametric representation of the line L. Lines The vector equation r = r0 + tv is short for x = x0 + tv1, y = y0 + tv2, z = z0 + tv3. If v1, v2, v3 6= 0, then we obtain the symmetric equations of L: x − x0 y − y0 z − z0 t= = = . v1 v2 v3 Exercise 7. Find the symmetric equations of the line L passing through h1, 2, −3i of direction h−1, −1, −1i. Exercise 8. Find the symmetric equations of the line L passing through h4, 1, 0i of direction h0, −1, −1i. Planes A plane H is determined by a point P0 = P0(x0, y0, z0) and a normal vector n. That, the position vector r for any other point P on H fulfills that n · (r − r0) = 0. In coordinates, with n = ha, b, ci, ax + by + cz = n · r0 or a(x − x0) + b(y − y0) + b(z − z0) = 0. Planes Any two non-parallel planes H1 and H2 intersect in a line L. Exercise 9. Determine a parametric representation of the intersection of the two planes defined by 2x + 3y + 2z = 0 and 4x − y − z = 0. Exercise 10. Determine the symmetric equations of the intersection of the two planes defined by x + y − z = 8 and 4x − 2y + z = 2. Exercises Exercise 11. Determine the distance from the point A(2, 4, 5) to the plane defined by x + 2y + 3z = 0