NAME: PERIOD: DATE: MATH ANALYSIS 2 MR. MELLINA CHAPTER 12: VECTORS & DETERMINANTS Sections: v 12.1 Geometric Representation of Vectors v 12.2 Algebraic Representation of Vectors v 12.3 Vector and Parametric Equations: Motion in a Plane v 12.4 Parallel and Perpendicular Vectors v 12.5 Vectors in Three Dimensions v 12.6 Vectors and Planes v 12.7 Determinants v 12.9 Determinants and Vectors in Three Dimensions HW Sets Set A (Section 12.1) Pages 423-425, # s 2-14 even, 18. Set B (Section 12.2) Pages 429 & 420, # s 2-24 even. Set C (Section 12.3) Pages 435 & 436, # s 2-22 even. Set D (Section 12.4) Pages 444 & 445, # s 2-24 even. Set E (Section 12.5) Pages 450 & 451, # s 2-28 even. Set F (Section 12.6) Page 455 & 456, # s 2-24 even, 28. Set G (Section 12.7) Page 460 & 461, # s 2-12 even. Set H (Section 12.8) Page 463 & 464, # s 2-20 even. Set I (Section 12.9) Pages 466 & 467, # s 2-14 even. 1
12.1 GEOMETRIC REPRESENTATION OF VECTORS (PAGE 418) Objective: To perform basic operations on vectors. 12.1 Warm Up! Suppose point X is directly east of point Y. a. Give the bearing of X from Y. b. Give the bearing of Y from X. Two ships, A and B, leave port at the same time. Ship A proceeds at 12 knots on a course of 040, while ship B proceeds at 9 knots on a course of 115. After 2 hours, ship A loses power and radios for help. c. Draw a diagram and use x to represent how far ship B must travel to reach ship A and then label the course as y. Vectors Quantities that are described by a and a (size). Ex # 1: Force is a vector quantity because to describe force, you must specify the direction in which it acts and its strength. Ex #2: Velocity is described by its direction and speed. Two airplanes heading northeast at 700 knots and one airplane heading east at 700 knots. Magnitude Denoted with the absolute value of v, v. Ex: v =, w = Example 1 Draw a Vector from point A going south to point B, AB that has a magnitude of 5 units. From point B, draw another vector going south east to point C, BC that has a magnitude of 6 units. Draw and name the vector from point A to point C. 2
Vector Addition Vector v is from A to B, v = AB ''''' AC ''''' is said to be the of AB ''''' '''''' + '''''' = '''''' Vector addition is a commutative operation (Order doesn t matter) The diagonal of the Parallelogram is the sum (resultant of the forces) One Newton (N) is the force needed to bring a 1kg mass to a speed of 1 meter per second in one second. **Vectors can in space** Example 2 a. Make a scale drawing showing a force of 20 N pulling an object east and another force of 10 N pulling the object in the compass direction 150. Draw the resultant force vector. Then use trig to find the magnitude (to the nearest hundredth) and the direction (to the nearest tenth of a degree) of the resultant. 3
b. Make a diagram showing the result of sailing a ship 3 miles on a course of 040 followed by sailing it 8 miles on a course of 100. Find the exact distance of the ship from its starting point. Vector Subtraction -The of a vector v, (-v) has the same length as v but the direction. -The sum of v and v is the vector 0 (a point) -Vectors can be subtracted Ex: v w = v + (-w) Multiples of a Vector -The vector sum v + v is abbreviated as 2v. (v + v + v = 3v etc) -if k is a real number, then kv is the vector with the same direction as v, but with the k value k times as large. -if k <, then kv has the same direction as v and has an absolute value k times as large. -if k, then -. is defined to be equal to the vector /. v. Example 3 Sketch a. 2u + 3v b. u 2v 4
Scalars -Real numbers are -k v is called scalar multiplication. -The distributive and associative laws also apply to scalars. Example 4 Quadrilateral ABCD is the parallelogram shown below. Tell whether each of the following is true or false. a. BC + BA = BD b. BC + BA = BD c. AO = AC d. AB + CD = 0 e. AO = OC f. AO = / 3 AC g. AB + BC + CD = AD h. AB + BC + CD = AD Example 5: Independent Work ** Use the same diagram as example 4 to complete the statements. a. AD = b. AD = c. / 3 BD = d. 2AO = e. AB + AD = f. AD + DC + CB = g. AO DO = AO + = h. BC BD = BC + = 5
Example 6: Independent Work * Sketch the following vectors a. Sketch an arrow representing a velocity v of 100 mi/h to the northwest. b. Sketch 2v and tell what velocity it represents. c. Sketch v and tell what velocity it represents. Example 7: Independent Work * Draw rectangle PQRS with PQ = 4, QR = 3. Complete a. PQ + QR = b. PQ + QR = c. PQ + RS = Example 8: Independent Work * Sketch u + v, u + 2v, u + 3v, and u v. 6
Example 9: Independent Work ** A ship travels 200 km west from port and then 240 km due south before it is disabled. Illustrate this in a vector diagram. Use trigonometry to find the course that a rescue ship must take from port in order to reach the disabled ship. Example 10: Independent Work ** Make a diagram that illustrates a force of 8 N north and force of 6 N west acting on a body. Illustrate the resultant sum of these two forces and estimate its strength. Then, using trigonometry, determine the approximate direction (as a number of degrees west of north) of this force. Example 11: Independent Work *** Make a vector diagram showing an airplane heading southwest at 600 knots and encountering a wind blowing from the west. Show the plane s resultant velocity when the wind blows at: a. 30 knots b. 60 knots 7
Example 12: Independent Work *** F / is a force of 3 N pulling an object north and F 3 is a force of 5 N pulling the object in the compass direction 060. a. Sketch both force vectors b. Sketch the resultant force vector F / + F 3 and use the law of cosines and the law of sines to find its magnitude and direction. c. Give the magnitude and direction of the force F > such that F / + F 3 + F > = 0 8
12.2 ALGEBRAIC REPRESENTATION OF VECTORS (PAGE 412) Objective: To use coordinates to perform vector operations. 12.2 Warm Up! Find AB and the slope of AB a. A(-4, 3), B(3, 2) b. A(r, s), B(s, r) Representing Vectors Algebraically The vector v consists of a 2 change in the -direction and -3 change in the -direction. 2 and -3 are called the of v. v = (2, -3) (component form) AB ''''' = (, ) BAB ''''' B = C( ) 3 + ( ) 3 Example 1 Given A(4, 2), B(9, -1) a. Express AB in component form b. Find AB 9
Rectangular Coordinates (a, b) and Polar Coordinates (r, θ) These two coordinates tie together 3 important ideas 1. Point 2. Vector 3. Complex number Point (a, b) Point (r, θ) Vector OP ''''' : (, ) Vector OP ''''' : (, ) Complex #: z = Complex #: z = Example 2 A force of 10 N acts on at an angle of 130 with the positive x-axis. Find F in component form. Example 3 Let u = (1, -3) and v = (2, 5), find: a. u + v b. u v c. 2u 3v 10
Example 4 Let A(0, 4) and B(6, 1). Find: a. The coordinates of point P that is 3 of the way from A to B. > b. The coordinates of point Q that is D of the way from A to B. E Example 5: Independent work * Plot points A and B. Give the component form of AB and find AB a. A(1, -2), B(3, -2) b. A(-3, -5), B(-5, 1) Example 6: Independent work * Polar coordinates of point P are given and O is the origin. Draw vector OP and give its component form. a. P(6, 72 ) b. P 2, FG > 11
Example 7: Independent work ** Let u = (3, 1), v = (-8, 4), and w = (-6, -2). Calculate each expression. a. u + v b. u v c. 3u + w d. 3u + w e. u + / 3 w f. u + / 3 w g. u + / 3 w Example 8: Independent work ** Find the coordinates of the point P described a. A(0, 0), B(6, 3); / of the way from A to B. 3 b. A(7, -2), B(2, 8); F of the way from A to B. D c. A(-7, -4), B(-1, -1); > of the way from A to B. D 12
Example 9: Independent work *** Suppose that you pull a child in a wagon by pulling a rope that makes a 60 angle with the ground. a. If the pulling force F is 40 lb in the direction of the rope, give the horizontal and vertical components of the force. b. Which component, horizontal or vertical, moves the wagon along the ground? 13
12.3 VECTOR AND PARAMETRIC EQUATIONS: MOTION IN A PLANE (PAGE 433) Objective: To use vector and parametric equations to describe motion in the plane. 12.3 Warm Up! Find all the points of intersection of the given circle and line. a. x 4 3 + y + 1 3 = 20 x + 3y = 11 b. What would need to be included in the equation if you were asked when do they intersect? Vector and Parametric Equations of Lines Let P be any point on line AB. OP ''''' = OA ''''' + -This is called a equation of line AB and AB ''''' is called the of the line. Example 1 Find a vector equation of the line through points A(3, 4), B(5, 5). 14
Representing t as Time If t is time in a vector equation, you can think of P as moving along the line, taking various positions at various times t. Vector Equation for an Object Moving with Constant Velocity Example 2 Suppose an object which is moving with constant velocity is at point A(5, 3) when the time t = 0 seconds and at point B(-4, 15) when the time t = 3 seconds. a. Find the velocity and speed of the object. b. Find a vector equation that describes the motion of the object. Parametric Equations We can derive 2 equations that give the x-coordinates and y-coordinates of the moving object at time t. (Parametric equation where t is the ) Vector Equation: Parametric Equation: Parametric equations are useful in finding the and at which an object moving with constant velocity crosses a curve whose equation is known. 15
Example 3 An object moves along a line in such a way that its x- and y-coordinates at time t are x = 1 t and y = 1 + 2t. When and where does the object cross the circle? Example 4: A line has the vector equation (x, y) = (2, -5) + t(1, 3). a. Name three points on the line b. Find a direction vector of the line c. Give a pair of parametric equations of the line. Example 5: An object moves with constant velocity along a line from A(-3, 1) at time t = 0 through B(5, 7) at time t = 2. Give a vector equation of line AB. 16
Example 6: Independent Work * Find the vector and parametric equations for each specified line. a. The line through (1, 5) with direction vector (2, -1) b. The line through (1, 0) and (3, -4) c. The horizontal line through π, e Example 7: Independent Work * A point moves in the plane so that its position P(x, y) at time t is given by the vector equation: (x, y) = (1, 4) + t(3, -2). a. Graph the point s position at the times t = 0, 1, 2, 3, -1, -2, -3 b. Find the velocity and speed of the moving point c. Find the parametric equations of the moving point Example 8: Independent Work ** A line has a vector equation (x, y) = (3, 2) + t(2, 4). Give a pair of parametric equations and a Cartesian equation of the line. 17
Example 9: Independent Work ** a. A line has direction vector (2, 3). What is the slope of the line? b. A line has a direction vector (4, 6). What is the slope of the line? c. Explain why the following lines are parallel: (x, y) = (8, 1) + r(2, 3) and (x, y) = (2, 5) + s(4, 6) d. Find a vector equation of the line through (7, 9) and parallel to these lines. Example 10: Independent Work *** An object moves with constant velocity so that its position at time t is (x, y) = (1, 1) + t(-1, 1). When and where does the object cross the circle x 1 3 + y 3 = 5? 18
12.4 PARALLEL AND PERPENDICULAR VECTORS; DOT PRODUCT (PAGE 441) Objective: To define and apply the dot product. 12.4 Warm Up! Give the component form of AB and find AB a. A(-1, -2), B(-7, 6) b. if C(0, 0) what is the angle formed by AB and AC? Parallel and Perpendicular Vectors: Algebraically vs. Geometrically Parallel Geometrically: 2 vectors are if the lines that contain them are. Algebraically: if v 3 =, for some real number k, thenv / & v 3 are. Perpendicular (Orthogonal) Geometrically: 2 vectors are perpendicular if the lines that contain them are. Algebraically: If R S T U T V R S W U W V = To show to vectors are perpendicular show that x / x 3 + y / y 3 =. The Dot Product (also called the scalar product) If v / = (x /, y / ) and v 3 = (x 3, y 3 ), then the dot product of the vectors v / & v 3 denoted v / v 3 is: v / v 3 = Example 1 If u = (3, -6), v = (4, 2), and w = (-12, -6). Find the following and show that u and v are perpendicular and that v and w are parallel. a. u v b. v w 19
Properties of the Dot Product The Angle Between Two Vectors 1. u v 2. u u cos θ = where 0 θ 180 3. k(u v) = 4. u (v + w) = Example 2 Find the measure θ, between u = (1, 2) and v = (-3, 1) Example 3: In ΔPQR, P = (2, 1), Q = (4, 7), R(-2, 4). Find m < P to the nearest tenth. Example 4: Identify which of the following three vectors are parallel and perpendicular and provide a statement justifying how you know. a. u = (4, -6), v = (-2, 3), w = (9, 6) 20
Example 5: Independent work * If u = (3, 4), v = (-2, 2), find: a. u v b. 2 u v c. 2u v d. u 2 v e. u f. u 3 g. u u Example 6: Independent work ** Verify that u v + w = u v + u w for the given vectors u, v, and w. a. u = (-2, 5), v = (1, 3), w = (-1, 2) 21
12.5 VECTORS IN THREE DIMENSIONS (PAGE 446) Objective: To extend vectors to three dimensions and to apply them. 12.5 Warm Up! Find the midpoint of RS. a. R(0, -7), S(-6, 3) b. Plot the point P(2, 3, 4) on the graph in the box below. Vectors in 3-Dimenstions xy-coordinate plane: the plane containing the - and -axis. yz-coordinate plane: the plane containing the - and -axis. xz-coordinate plane: the plane containing the - and -axis. : the 8 regions created by the 3-D planes A(,, ) In General B(,, ) C(,, ) AB ''''' = C(,, ) The Distance and Midpoint Theorems in 3-Dimensions BAB ''''' B = C( ) 3 + ( ) 3 + ( ) 3 The midpoint of AB cccc = R d, d, d V 22
Example 1 A sphere has points A(8, -2, 3) and B(4, 0, 7) as endpoints of a diameter. Find: a. Find the center C and radius r of the sphere b. Find an equation of the sphere The Line Containing (x 0, y 0, z 0 ) with Direction (a, b, c) Vector Equation: (x, y, z) = (,, )+ t(,, ) Parametric Equation: x = y = z = Example 2 Find the vector and parametric equations of the line containing A(2, 3, 1) and B(5, 4, 6) 23
Example 3: Find the vector equation through (1, 5, -2) and parallel to the line L with equation (x, y, z) = (8, 0, 1) + t(4, 3, 2). Example 4: Find θ between (4, -5, 3) and (7, 0, -1) Example 5: Independent work * Find the length and midpoint of AB. a. A = (2, 5, -3), B = (0, 3, 1) Example 6: Independent work * Simplify the expression a. (3, 8, -2) + 2(4, -1, 2) b. (1, -8, 6) (5, 2, 1) c. 3, 5, 1 24
Example 7: Independent work * Are vectors (3, -7, 1) and (6, 3, 3) perpendicular? Example 8: Independent work ** Find an equation of the sphere with radius 2 and center at: a. the origin b. (3, -1, 2) Example 9: Independent work ** Find the angle between (8, 6, 0) and (2, -1, 2) to the nearest tenth of a degree. 25
12.6 VECTORS AND PLANES (PAGE 452) Objective: Vectors and Planes 12.6 Warm Up! Find the missing coordinate of each point, given that the points lie on the graph of 2x 3y 6z = -12 a. (, 0, 0) b (0,, 0) c. (0, 0, ) What do these points represent in the 3D plane? The Cartesian Equation of a Plane If (a, b, c) is a nonzero vector perpendicular to a plane at the point (x 0, y 0, z 0 ), then an equation of the plane is: ax + by + cz = d, where d = ax n + by n + cz n. If a vector is perpendicular to a plane containing points, then the vector is also perpendicular to those vectors and all other vectors in the plane. Example 1 Vector (3, 4, -2) is perpendicular to a plane that contains A(0, 1, 2). Find the equation of the plane. Example 2 If A = (1, 0, 2) and B = (3, -4, 6), find a Cartesian equation of the plane perpendicular to AB at its midpoint. 26
Example 3: Find the x-, y-, and z-intercepts of the plane 3x + y + 2z = 6 Example 4: A rectangular box is shown, find an equation of the plane that contains: a. the top of the box b. the bottom of the box c. the front of the box d. the right side of the box Example 5 Sketch each plane whose equation is given a. 2x + 3y + 6z = 12 b. z = 2 Example 6: Independent work * Find a vector perpendicular to the plane whose equation is given. a. 3x + 4y + 6z = 12 b. x + y = 4 c. z = 1 27
Example 7: Independent work ** Find a Cartesian equation of the plane described. a. Vector (2, 3, 5) is perpendicular to the plane contains point A(3, 1, 7) Example 8: Independent work ** Consider the points A(2, 2, 2) and B(4, 6, 8) a. find a Cartesian equation of the plane that is perpendicular to AB at its midpoint M. b. Show that point P(2, 0, 8) satisfies your answer to part (a) c. Show that PA = PB 28
Example 9: Independent work *** Find an equation of the plane tangent to the sphere provided at the given point. a. x 1 3 + y 1 3 + z 1 3 = 49 P(7, -1, 4) 29
12.7 DETERMINANTS (PAGE 458) Objective: To define and evaluate determinants. 12.7 Warm Up! Let a b represent the expression ad bc. Evaluate: c d a. 3 2 5 7 b 5 7 3 2 c. Can the value of a b c d be zero? Determinants The expression q a / a 3 b / b q is called a determinant (2 and 2 ) 3 (Subtract the product of the diagonals) To find the value of this determinant: q a / b 3 q q b / a 3 q a / and b 3 are elements of the diagonal while a 3 and b / are elements of the diagonal. Example 1 Find the value of the determinant a. 3 4 2 7 b. 4 1 2 0 30
3 x 3 Determinants a / a 3 a > The general 3 x 3 determinant is shown at the right: sb / c / b 3 c 3 b > s c > Each, or number, in this determinant is associated with a 2 x 2 determinant called its. The minor of an element is the determinant that remains when you cross out the and containing that element. Ex: The minor element a 3 is t b / b > t c / c > You can evaluate a 3 x 3 determinant by expanding by the minors of the elements in any row or column. However, you have to consider the position of each element in the original determinant and find its corresponding position on a checkerboard pattern to know its sign. Example 2 Evaluate using the conditions specified a. 5 2 8 3 4 1 7 1 6 by expanding minors of the first row. b. 5 2 8 3 4 1 7 1 6 by expanding minors of the second column. 31
Example 3: Independent work * Evaluate each determinant a. 2 7 8 4 b. 25 125 75 250 Example 4: Independent work * Show that a b a c b d = c d c d Example 5: Independent work ** Evaluate each determinant a. 4 7 3 2 0 0 5 1 6 b. 1 3 4 0 1 1 5 2 3 32
Example 6: Independent work **** Evaluate each determinant a. 2 4 7 0 5 1 2 0 1 6 0 4 3 2 3 2 33
12.8 APPLICATIONS OF DETERMINANTS (PAGE 461) Objective: To use determinants to solve algebraic and geometric problems. 12.8 Warm Up! Evaluate each determinant a a 2 1 3 a. b 1 a 3 2 1 1 3 2 Applications of Determinants Determinants can be used to solve n equations in n variables. This method is called. (A special calculator is needed for n 4) Example 1 Use Cramer s rule to solve: 3x + 5y = 7 4x + 9y = 11 34
Example 2 Solve the system of equations: 3x y + 2z = 4 2x + 3y z = 14 7x 4y + 3z = -4 35
Geometric Applications of Determinants We can use determinants to find area and volume, as shown below. Area Volume The area of the parallelogram with The volume of a parallelepiped with sides determined by v / & v 3 is the edges determined by v /, v 3, and v > is absolute value of: the absolute value of: t a a / b / b / c / / t sa a 3 b 3 b 3 c 3 s 3 a > b > c > Example 3: Find the area of the triangle with vertices P(1, 2), Q(3, 6), and R(6, 1). Example 4: Independent work Show each system of equations by using Cramer s rule. a. 5x 4y = 1 b. 3x + 2y = 1 3x + 2y = 5 2x y = 4 36
c. ax + by = 1 d. 9x 6y = 3 bx + ay = 1 6x 4y = 10 Example 5: Independent work Find the area of each figure, given points P(4, 3), Q(7, -1), R(2, 3), S(-3, 6), T(-5, 4), and V(-2, -5). a. ΔPQR b. with sides PR, PS Example 6: Independent work If the area of ΔLMN is zero, what can you conclude about points L, M, and N? 37
Example 7: Independent work Solve each system of equations by using Cramer s rule. a. x 2y + 3z = 2 2x 3y + z = 1 3x y + 2z = 9 b. 3x 2y + z = 7 2x + y 3z = 1 x + 2y + 2z = 4 38
12.9 DETERMINANTS AND VECTORS IN THREE DIMENSIONS (PAGE 465) 12.9 Warm Up! Find a Cartesian equation of the plane that contains the point P(1, -1, 2) and to which the vector (2, 3, 1) is perpendicular. The unit vectors in the positive x, y, and z directions are often denoted by the symbols i, j, and k. Ex: Point P(3, 5, 6) OP ''''' = This representation of a vector is useful in defining a new vector operation, the product of the two vectors. Cross Product of 2 Vectors If v / = (a /, b /, c / ) and v 3 = (a 3, b 3, c 3 ), then the cross product of v / and v 3, denoted v / v 3 is found by evaluating: i j k sa / b / c / s a 3 b 3 c 3 Example 1 Let v / = 2, 3, 4 and v 3 = 1, 0, 5, then find a. v / v 3 39
Properties of the Cross Product 1. u v is perpendicular to u and to v. 2. v u = (u v) that is, v u and u v have opposite directions. 3. u v = u v sin θ, where θ is the angle between u and v. Geometrically, the magnitude of u v is the area of the parallelogram formed by u and v. 4. u (v + w) = (u v) + (u w). 5. u is to v if and only if u v = 0. Example 2 Use the points P(1, 0, 3), Q(2, 5, 0), and R(3, 1, 4). a. Find a nonzero vector perpendicular to the plane containing the points provided. b. Find an equation of the plane containing P, Q, and R. c. Find the area of the parallelogram with sides formed by PQ and PR. 40
Example 3: Tell whether each of the following is a vector or a scalar. a. u v b. u v c. u + v d. u u + v e. u u + v f. u u v Example 4: Independent work * Let u = (4, 0, 1), v = (5, -1, 0), and w = (-3, 1, -2). a. Calculate v u and u v. Do your results agree with property 2? b. Verify that u v is perpendicular to u and to v. c. Find the area of the parallelogram determined by u and v. d. Verify that u v + w = u v + u w 41
Example 5: Independent work ** Use the following points: P(1, 1, 0), Q(-1, 0, 2), R(2, 1, 1). a. Find a vector perpendicular to the plane determined by P, Q, and R. b. Find a Cartesian equation of the plane determined by P, Q, and R. c. Find the area of ΔPQR 42
Example 6: Independent work ** Angle θ is between vectors u = (1, 2, 2) and v = (4, 3, 0) a. Find sin θ by using property 3 of the cross product. b. Find cos θ by using property 5 of the dot product. c. Verify that sin 3 θ + cos 3 θ = 1 43
12.1 & 12.2 Quiz Review 1. Sketch an arrow representing a ship s trip w of 200 mi on a course of 200. 2. Sketch 3w and tell what trip 3. Sketch -1.5w and tell what trip it represents it represents 4. An object is pulled due South by a force F / of 5N, and due east by a force F 3 of 12N. Find the direction and magnitude of F > = F / + F 3. 5. An airplane has a velocity of 400 mi/hr southwest. A 50 mi/hr wind is blowing from the west. Find the result speed and direction of the plane. 44
6. Given A(2, -4) and B(1, 1). FindAB in component form. a. Find AB b. Give polar coordinates of AB 7. Let u = (-5, 2), v = (-3, -6) a. Find 2u + v b. 2u + v c. 2u + v d. u v u v e. Find the polar form of v 45
8. Given A(-7, -4), B(-1, -1); Find the point > of the way from A to B. D 9. Given A(7, -2), B(2, 8); Find the point F of the way from A to B. D 10. Find the real numbers r and s such that r(1, 2) + s(3, 0) = (-1, 4). Illustrate your solution with a diagram. 11. Find a vector of length 1 in the same direction as (-2, -6). 46
12.3-12.6 Quiz Review 1. A point moves along a line starting at (4, -2) and reaches (0, 7) when t = 3. a. Give a vector equation for. b. Give the parametric equations for this line. this line. 2. An object moves with constant velocity along the vector (x, y) = (-1, 3) + t(2, 8). At what points does it cross the curve y = x 3 38? 3. Given vectors u = (3, -1) and v = (-5, 3), find the following: a. u v b. v c. 2u + v d. a vector parallel to u e. A vector perpendicular to v f. the angle between vectors u and v. 47
4. Given points A(1, 4, 0) and B(-2, 1, 5), find the following: a. midpoint of AB b. Distance between A and B c. A vector equation for AB (Position at A when t = 0 and at B when t = 1) d. AB 5. The equation of a plane is 3x 7y z = 14. Find the coordinates of the x, y, and z- intercepts. 6. Find the equation for the plane that contains the point (-2, 8, 3) and is perpendicular to the vector from A(2, 0, 3) to B(3, 5, 1). 48
12.3-12.6 Review Answers 49
Additional Review Problems: Sections 12.3-12.6 50
Additional Review Problems Continued: Sections 12.3-12.6 51
Additional Review Problem Answers 52
Additional Review Problems: Sections 12.5 & 12.6 53
Additional Review Problems Continued: Sections 12.5 & 12.6 54
Additional Review Problem Answers 55