Monday Tuesday Block Friday 13 22/ End of 9-wks Pep-Rally Operations Vectors Two Vectors

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1 Name: Period: Pre-Cal AB: Unit 6: Vectors Monday Tuesday Block Friday /16 PSAT/ASVAB 17 Pep Rally No School Solving Trig Equations TEST Vectors Intro /23 24 End of 9-wks Pep-Rally Operations Vectors Two Vectors Applications QUIZ 27 Start Next Unit 28 Vectors TEST Lesson #1:Vectors Introduction I can graph vectors to find the resultant vector. find the component and magnitude of a vector. I. Vectors A. Definition: Quantities that have and B. Initial Point: point (P) C. point the final point (Q). Found by D. Magnitude the length of the vector ( PQ) E. Written either as or. F. Component of a vector - a, b II. Alternate Vector Notation linear combination of i and j A. i = <1,0> and j = <0,1> B. v = < a, b > = ai + bj C. This form is called the resultant vector (v = ai + bj) D. The sum of all the forces acting on an object is called the resultant force. II. Graphing Vectors A. Vectors can move anywhere on the plane. You must only keep the (length) and (angle measure) B. When adding put the tail of the 2 nd vector on the of the first vector. Connect the original to the new. C. To subtract, go in the direction of the 2 nd vector making it to. IV. Model Problems Guided Practice On Your Own Ex1: Given u and v, find u + v, u v and 2u + 3v. Ex2: Given u and v, find u + v, u v and 2u + 3v. u v u 2 2 u = a + b v

2 Ex3: Find the component and magnitude if P = (-2, 6) and Q = (4, -3) Ex4: Find the component and magnitude if P = (0, 8) and Q = (-4, -7) Ex5: Let u = 5, 2 and v = 3,1, Find u + v by graphing. Ex6: Find the magnitude of the component vector from the left. Practice #1 Find the magnitude of the vector PQ 1) P = (2, 3) Q = ( 5, 9) 2) P = (-3, 5) Q = (7,-11) 3) P (30, 12) Q (25, 5) Copy each vector and find the indicated resultant vector. u v 15) u + v 16) 3u 17) v u 18) u v 19) - 2v 20) 2v + u

3 Lesson #2: Vector Operations I can Find the unit vector in the same direction as the original vector. - perform scalar multiplication of vectors, vector addition, and vector subtraction. - Write vectors in an alternate notation. - Find the resultant vector I. Operations Using Mathematics A. Scalar Multiplication Multiply the vector by a quantity a. B. Addition Add the x components and add the y components C. Subtraction the negative then add. II. Unit Vectors A. Unit Vector a vector with length (magnitude) 1 B: Find a unit vector 1. Find the length of the vector 2. Divide each component by the length of the vector. C. A unit vector is in the same direction as the original vector IV. Model Problems Ex1: If v = 3,1, find 3v and -2v Ex2: Let u = 5, 2 and v = 3,1, find u + v Ex3: Let u = 5, 2 and v = 3,1, find 4u 3v Ex4: Find a unit vector u with the same direction as the vector = 5,12 Ex5: Find a unit vector u with the same direction as the vector = 2,3 Ex6: Find the resultant vector if P = (-2, 6) and Q = (4, -3) Ex7: Find the resultant vector if P = (4, -3) and Q = (-5, 2) Ex8: u = 2i 6j and v = -5i + 2j Find u+v and 2u - v Ex9: u = 3i + 4j and v = -2i - 6j Find u-v and 2u +4v

4 Practice #2: Find u + v, u v, and 3u 2v. 1) u = 2, 4 v = 6,1 2) u = 4, 0 v = 1, 3 3) u = 2 2, 4 v = 2, 1 Let u = 8, 4, v = 3,1, and w = 6, 2, find: 4) u + v 5) u v 6) 3u + v 7) v + w 8) 2(v w) 9) u + ½ w 10) -2(w + 2u) 11) 7 v 2 u 6 3 Find a unit vector in the same direction as the given vector. 12) = 4, 5 13) = 7,8 14) = 5,10 15) = ) = 1,3 17) = 6 2 Find the resultant vector given the two end points. 18) = 6,5, = 3,8 19) = 2,12, = 1, 10 Given =3 7, =4 2 = Perform operations with linear combinations. 20) + 21) 22) ) ) ) +2 +5

5 Lesson #3: Two Vectors I can Find the dot product of two vectors. - Find the cross product of two vectors. - Find the angle between two vectors - Determine whether two vectors are parallel, perpendicular, or neither. I. Operation with vectors A. Dot Product: If =, and =, then = B: Cross Product: If =, and =, then = II. Properties of Dot Product If u, v, and w are vectors, and k is a real number, then: A. = B. = C. + = D. = E. 0 = II. Angle between vectors A. If is the angle between the nonzero vectors u and v, cos _ = III. Parallel, perpendicular (orthogonal) or neither A. Vectors u and v are parallel when =0 B. Vectors u and v are orthogonal when =0 IV. Model Problems Ex 1: Given = 5,3 = 2,6 Find and Ex 2: Given =4 2 = 3 Find and Ex 3: Find the angle between the vectors = 3,1 = 5,2 Ex 4: Find the angle between the vectors =4 3 = +2 Ex 5: Given = 1,3, = 1,2 and = 2, 5 Find + Ex 6: Given = 1,3, = 1,2 and = 2, 5 Find + + Ex 7: Determine whether vectors u and v are parallel, orthogonal or neither. = 2, 6 = 9,3 Ex 8: Determine whether vectors u and v are parallel, orthogonal or neither. =3 6 = 5 +3

6 Ex 9: Determine whether vectors u and v are parallel, orthogonal or neither. = 3,2 = 6, 4 Ex 10: Determine whether vectors u and v are parallel, orthogonal or neither. = 9, 3 = 3,1 Practice #3: 1 4 Find and 1. = 3,4 = 5,2 2. = 1,6 = 4, 3. =2 + = 3 4. = = Find the angle between the vectors. 5. = 2,4 = 0, 5 6. = 2, 3 = 1,0 7. =2 = =3 5 = Given = 1,3, = 1,2 and = 2, Determine whether vectors u and v are parallel, orthogonal or neither. 15. = 2,6 = 3, = 5,3 = 2,6 17. = 9, 6 = 6,4 18. = 1,2 = 2, = 2, 2 = 5,8 20. = 6, 4 = 2,3

7 Lesson #4: Vector Applications I can Find the component of the direction angle. - Write vectors in an alternate notation. - Find the resultant vector I. Components of the Direction Angle A. If v = =, then a = and b = where is the direction angle of v. B. The angle = Ex 1: Find the component form of the vector that represents the velocity of an air plane at the instant its wheels leave the ground, if the plan is going 60 miles per hour and the body of the plan makes a 7 angle with the horizontal. Ex 2: Find the direction angle of each vector u = 5i + 13j and v = -10i + 7j Ex 3: An object at the origin is acted upon by two forces. A 150-pound force makes an angle of 20 with the positive x-axis, and the other force of 100-pounds makes an angle of 70 with the positive x-axis. Find the direction and magnitude of the resultant force. Ex 4: A 200-pound box lies on a ramp that makes an angle of 24 with the horizontal. A rope is tied to the box from a post at the top of the ramp to keep it in position. Ignoring friction, how much force is being exerted on the rope by the box? Ex 5: An airplane is flying in the direction 50 with an air speed of 300 miles per hour. If there was no wind, the course of the airplane would be 50. However, there is a 35 mph wind from the direction 120. Find the course and ground speed of the plane.

8 Practice #4: Find the magnitude and direction angle of each vector. 1) = 4,4 2) =4 8 3) = ) An object at the origin is acted upon by two forces u = 30 pounds with a direction angle of 0 and v= 90 pounds with a direction angle of 60. Find the direction and magnitude of the resultant force. 5) An object at the origin is acted upon by two forces u = 6 pounds with a direction angle of 45 and v= 6 pounds with a direction angle of 120. Find the direction and magnitude of the resultant force. Find the course and ground speed of the plane under the given conditions. 6) air speed 250 miles per hour in the direction of 60 ; wind speed 40 miles per hour from the direction 330 7) air speed 400 miles per hour in the direction of 150 ; wind speed 30 miles per hour from the direction 330 8) air speed 300 miles per hour in the direction of 300 ; wind speed 50 miles per hour in the direction 30 9) A force of 500 pounds is needed to pull a cart up a ramp that makes a 15 angle with the ground. Assuming that no friction is involved, find the weight of the cart.