Chapter 12 Review Vector. MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 1 / 30

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1 Chapter 12 Review Vector MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 1 / 30

2 iclicker 1: Let v = PQ where P = ( 2, 5) and Q = (1, 2). Which of the following vectors with the given initial and terminal points are equivalent to v? (A) ( 3, 3) and (0, 4) (C) ( 1, 2) and (2, 5) (B) (0, 0) and (3, 7) (D) (4, 5) and (1, 4) MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 2 / 30

3 Example 1: Find a vector in the opposite direction of v = 10, 1, 6 with magnitude 20. MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 3 / 30

4 MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 4 / 30

5 MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 5 / 30

6 MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 6 / 30

7 MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 7 / 30

8 MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 8 / 30

9 Projections (a.k.a. Parallel Projections) The vector projection of v onto u is the vector proj u ( v) given by proj u ( v) = ( ) u v u 2 u MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 9 / 30

10 Projections (a.k.a. Parallel Projections) The vector projection of v onto u is the vector proj u ( v) given by proj u ( v) = ( ) u v u 2 u Practice Problems 1. Find the vector projections of b onto a. a = 4, 3, b = 6, 0 2. Show that the vector orth b (a) = a proj b ( a) is orthogonal to b. MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 9 / 30

11 iclicker 1 The value of the expression is (A) 0 (B) -4v.w (C) 4v.w (D) 8w.v v + w 2 v w 2 = MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 10 / 30

12 Example 9: Calculate the determinant for the following matrices: Example 3: Let u = 2, 5, 10 and v = 6, 3, 5. Compute u v and then verify that it is orthogonal to both u and v. Example 4: Find the two unit vectors orthogonal to both a = 3, 1, 1 and b = 1, 2, 1. MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 11 / 30

13 iclicker 2 Lets assume that u v = 1, 1, 0, u w = 0, 3, 1, v w = 2, 1, 1 Then the cross product of (3 u + 4 w) w is (A) < 0, 0, 0 > (B) < 0, 9, 3 > (C) < 1, 1, 0 > (D) < 6, 3, 3 > MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 12 / 30

14 Equation of a Plane The equations of the plane through P = (x 0, y 0, z 0 ) with normal vector n =< a, b, c > Vector forms: n.< x, y, z >=d Scalar forms: a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0 Where d = n. < a, b, c >. Example 4 Find an equation of the plane passing through the three points given. Also find the area of triangle formed by these points P = (4, 1, 1), Q = (1, 1, 3), R = (2, 1, 1) Example 5 Find the plane through (4, 1, 9) containing the line r(t) =< 1 + 2t, 4 + t, 3 + t >. MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 13 / 30

15 The dot product of two vectors v and u is the number given by v u = u v cos (θ) where θ is the small angle between the vectors v and u. Dot Product in Component Notation Suppose that v = a 1, b 1, c 1 and u = a 2, b 2, c 2. Practice Problems v u = a 1 a 2 + b 1 b 2 + c 1 c 2 2. Find a.b. a = 3, 1, 1/5 and b = 6, 2, Find the angle between a diagonal of a cube and one of its edges 4. Determine whether the given vectors are orthogonal, parallel, or neither. u = 3, 3, 3, v = 4, 4, 4 MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 14 / 30

16 Projections (a.k.a. Parallel Projections) The vector projection of v onto u is the vector proj u ( v) given by proj u ( v) = ( ) u v u 2 u MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 15 / 30

17 Projections (a.k.a. Parallel Projections) The vector projection of v onto u is the vector proj u ( v) given by proj u ( v) = ( ) u v u 2 u Practice Problems 1. Find the vector projections of b onto a. a = 4, 3, b = 6, 0 2. Show that the vector orth b (a) = a proj b ( a) is orthogonal to b. MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 15 / 30

18 Example 9: Calculate the determinant for the following matrices: Example 10: Let u = 2, 5, 10 and v = 6, 3, 5. Compute u v and then verify that it is orthogonal to both u and v. MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 16 / 30

19 Example 9: Calculate the determinant for the following matrices: Example 10: Let u = 2, 5, 10 and v = 6, 3, 5. Compute u v and then verify that it is orthogonal to both u and v. Example 11: Calculate the cross product assuming that u v = 1, 1, 0, u w = 0, 3, 1, v w = 2, 1, 1 (A) v u (B) w ( u + v) (C) (3 u + 4 w) w (D) ( u 2 v) ( u + 2 v) Example 12: Find the two unit vectors orthogonal to both a = 3, 1, 1 and b = 1, 2, 1. MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 16 / 30

20 Section 12.7 A Survey of Quadric Surfaces Conic Sections Quadric Surfaces MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 17 / 30

21 Review: Conic Sections In R 2, conic sections are curves satisfying an equation of the form Ax 2 + Bx + Cy 2 + Dy + E = 0 Ellipse ( ) x h 2 ( ) y k 2 + = 1 a b Hyperbola x 2 y 2 = a 2 y 2 x 2 = a 2 MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 18 / 30

22 In R 3, quadric surfaces are surfaces satisfying an equation of the form Ax 2 + By 2 + Cz 2 + Dxy + Exz + Fyz + ax + by + cz + d = 0 We classify these surfaces on the type of conic section obtained by intersecting the surface with planes parallel to the xy, xz, and yz planes. MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 19 / 30

23 In R 3, quadric surfaces are surfaces satisfying an equation of the form Ax 2 + By 2 + Cz 2 + Dxy + Exz + Fyz + ax + by + cz + d = 0 We classify these surfaces on the type of conic section obtained by intersecting the surface with planes parallel to the xy, xz, and yz planes. Conic Cylinders: Surfaces with an equation containing only 2 of the three variables. Cylinders open along the axis of the missing variable. MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 19 / 30

24 Ellipsoid: All plane intersections result in an ellipse. ( x a + ( y b + ( z c = 1 MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 20 / 30

25 Ellipsoid: All plane intersections result in an ellipse. ( x a + ( y b + ( z c = 1 Elliptic Paraboloid: Two intersections result in parabolas and the third results in an ellipse. ( x ( y z = + a b MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 20 / 30

26 Hyperboloid of One Sheet: Two intersections result in hyperbolas and the third results in an ellipse. ( x a + ( y b = ( z c + 1 MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 21 / 30

27 Cone: Two intersections result in hyperbolas and the third results in an ellipse. ( x a + ( y b = ( z c Intersecting z = 0 results in a single point. MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 22 / 30

28 Cone: Two intersections result in hyperbolas and the third results in an ellipse. ( x a + ( y b = ( z c Intersecting z = 0 results in a single point. Hyperboloid of Two Sheets: Two intersections result in hyperbolas and the third results in an ellipse. ( x a + ( y b = ( z c 1 For z in ( c, c), there are no (x, y) solutions to the equation. MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 22 / 30

29 Hyperbolic Paraboloid: Two intersections result in parabolas and the third results in a hyperbola. ( x ( y z = a b Intersecting z = 0 results in a single point. Along the z-axis, the parabola from yz-planes opens positively. Along the z-axis, the parabola from xz-planes opens negatively. MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 23 / 30

30 MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 24 / 30

31 Example 1: Sketch and identify the graph of 4x 2 y 2 + 2z = 0 MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 25 / 30

32 Example 1: Sketch and identify the graph of 4x 2 y 2 + 2z = 0 Solution: The equation defines a Hyperboloid of 2-Sheets. ( ) z 2 ( y x 2 + = MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 25 / 30

33 iclicker Question A Which of the following is graphed below? (A) z = (x 3 + (y 1 (B) x 3 = (y 1 + z 2 (C) y 1 = (x 3 + z 2 (D) (x 3 = (y 1 + z 2 MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 26 / 30

34 iclicker Question B Identify the graph of 9x 2 + 4y 2 + z 2 = 1. (A) Hyperboloid of 1-Sheet (C) Cone (B) Hyperboloid of 2-Sheets (D) Ellipsoid MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 27 / 30

35 iclicker Question C Identify the graph of x 2 y 2 + z 2 = 1. (A) Hyperboloid of 1-Sheet (C) Cone (B) Hyperboloid of 2-Sheets (D) Elliptical Paraboloid MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 28 / 30

36 iclicker Question D Identify the graph of x 2 y 2 + z 2 = 1. (A) Hyperbolic Paraboloid (C) Cone (B) Hyperboloid of 2-Sheets (D) Elliptical Paraboloid MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 29 / 30

37 iclicker Question E Identify the graph of y = x 2 z 2. (A) Hyperbolic Paraboloid (C) Cone (B) Ellipsoid (D) Elliptical Paraboloid MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 30 / 30

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