MA Lesson 20 Section 8.3 Vectors. We use PQ to denote the vector with initial point P and terminal point Q.
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1 We use PQ to denote the vector with initial point P and terminal point Q. The name of this vector is v. P v A vector is a force that has both magnitude and direction. The direction is indicated b the arrow at the terminal point. The magnitude is the length of the segment representing the vector. v v (magnitude of the vector v) represents the length of the vector. Q Ever vector has 2 components; a direction and a magnitude, which is represented b the length from P to the arrow. These are not absolute value bars. A vector that represents a pull or push of some tpe is a force vector. Eamples of force vectors include a car (magnitude is speed and direction is obvious) and a thrown ball (force would be determined b size or weight of ball and speed with which it was thrown). A single force that represents the combined forces of two combined vectors is a resultant force. An eample of a resultant force would be one car hitting another car. Each car has a force. Where the hit car ends up after being hit b the second car is a result of its own force and the force of the car that hit it. We use a parallelogram to determine the resultant force. The diagonal of the parallelogram represents the resultant force. a and b combined together make a resultant vector r represented b the diagonal of the parallelogram formed. a r b Parallelograms are reviewed on the here and the picture is shown on the net page. Adjacent angles of a parallelogram are supplementar, opposite angles are congruent. Opposite sides are parallel and congruent. Diagonals will not be congruent. 1
2 can be doubled, tripled, halved, etc. This new vector could be represented b multipling the original vector b a number m. If a vector is multiplied b 2, the result is a vector in the same direction, but twice the length. If a vector is multiplied b -2, the result is a vector in the opposite direction and twice the length. mv is a scalar multiple of the vector v. If m > 0 then it has the same direction as v. If m < 0 then it has the opposite direction as v. B placing a vector s initial point at the origin, the -plane is used to represent the vector. The numbers a 1 and a 2 are the components of vector < a 1, a 2 >. The direction of the vector is determined b graphing a 1 and a 2. The magnitude of the vector is the length of the segment. Vector d shown would have the components 10, 3. Components d (10, 3) is comparable to an ordered pair. The magnitude of vector d could be found b using the Pthagorean Thm. Magnitude of a vector: a = a = a a Addition of vectors: a 1 + b1, b2 = a1 + b1 + b2 Subtraction of vectors: a1 b1, b2 = a1 b1 b2 Scalar multiple of a vector: m a1 = ma1, ma2 2
3 E 1) Find the magnitude of the vector 2,5 E 2) Find the addition vector and subtraction vectors below. 1, 9 + 4, 2 6, 8 5, 3 When adding vectors the result would be the diagonal of the parallelogram resulting from the two given vectors. E 3) Find the components of the following vectors: 4 3, 7 and 3 2,1 The first would be a vector 4 times as long in the same direction. The second, a vector 3 times as long in the opposite direction. A unit vector is an vector with a length 1 unit. However, there are two special unit vectors that are 1 unit long from the origin on the positive -ais direction and on the positive -ais direction. Special vectors i = 1,0, j = 0,1 are unit vectors of magnitude 1. Using the special unit vectors above leads to a second wa to denote vectors. An alternate wa of denoting vectors: a = a1 = a1i + a2 j The i here is not the imaginar unit. E 4) Write each vector in the alternate wa. 4,5 = 0, 6 = 4i 2 j = 3
4 E 5) Sketch vectors a and b, then find and sketch 2a, b, a + b, a b, and 3a + 2b a = 3,4, b = 4, 2 a = 2i + 3 j, b = 4i 2 j a + b = a b = 3a + 2b = 2a b = a + b = a b = 3a + 2b = 2a b = Note: The vector a + b is the diagonal (resultant force) of a parallelogram formed b vectors a and b. Find the magnitude of a and the smallest positive angle θ from the positive -ais to the vector OP that corresponds to a. a = 2 3, 2 tanθ = 2 1 = = π θr = 6 π 7π θ = π + = 6 6 4
5 MA Lesson 20 Section 8.3 a = 2i + 3j The vectors a and b represent two forces acting at the same point, and θ is the smallest positive angle between a and b. Approimate the magnitude of the resultant force. a = 40 lb, b = 70 lb θ = 45 We will use the law of cosines. However, we need to complete the parallelogram to find the angle. a = 30 kg, b = 50 kg, θ = 150 5
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