Lesson_Text_1.3 - 1 Lesson 1.3 Vectors 1. Vector Addition...

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1 Lesson 1.3 Vectors 1. Vector Addition In lesson 2 you learned about vectors. Vectors are physical quantities that have both magnitude and direction, and can be represented by drawing an arrow with the length of the arrow proportional to the magnitude of the vector and arrow head representing the direction. The arrow end is the head of the vector and the other end is the tail. A vector can be drawn any where in a plane as long it has the same magnitude and direction. For convenience of adding vectors, we will classify all vectors as x-vector or y-vector. An x-vector is a vector in the x-direction either positive or negative. Similarly a y-vector is a vector in the y-direction either positive or negative. + = Fig. 1. Tw o positive x-vectors add to give a single x-vector. A B C Fig 1 above shows two positive x-vectors A and B adding to give a positive x-vector C . The magnitude of C = A + B . The result of adding two or more vectors is called the resultant vector . In fig. 1 C is the resultant of vectors A and B. A B C + = Fig. 2. C is the resultant of vectors A and B In fig. 2, A is a positive x-vector and B is a negative x-vector. Since the magnitude of A is greater than the magnitude of B , the resultant C is a positive x-vector which has a magnitude given by C = A B and it is in the direction of the larger vector. In a similar fashion, we can add y-vectors. Now we will add a y-vector to an x-vector. For example, Tom walked 5.0 m along the positive x-direction and then 3.0 m in the positive y-direction. What is the resultant displacement? The 5.0 m x-vector and the 3.0 m y- vector are shown in fig. 3a below.
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2 5.0 m 3.0 m 5.0 3.0 r e s u l t a n v c o Fig. 3a Fig. 3b A B C θ In order to add them, draw the 5.0 m vector first. In fig. 3b above, AB represents this vector. Draw the 3.0 m vector with its tail at the head of the 5.0 m vector. In the figure, BC is this vector. Now join AC and this is the resultant of the two vectors. Notice here that the direction of the resultant is from A to C as is indicated by the arrow. The figure ABC is a right triangle with AB and BC as its legs and AC the hypotenuse. The result of adding vector AB to vector BC is the vector AC . The magnitude of the resultant is the length of AC which can be calculated using AC 2 = AB 2 + BC 2 = 5 2 + 3 2 = 34. Therefore, AC = 5.8 m. The direction of the vector is obtained by measuring the angle BAC = θ . In the right triangle ABC, 6 . 0 5 3 tan = = = AB BC θ . Therefore, θ = tan -1 (0.6) = 31 o The resultant of the two vectors is 5.8 m at an angle 31 o measured with the positive x-axis. 2. Unit vectors i and j: If i is a unit vector in the x-direction, then 4 i represent a vector of magnitude 4 units in the positive x-direction. –2 i is a vector of magnitude 2 units in the negative x-direction. Adding these two vectors produces 4 i – 2 i = 2 i , a vector of 2 units in the positive x-direction. Similarly, if j is a unit vector in the y-direction, then 6 j represents a vector of magnitude 6 units in the
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This note was uploaded on 05/01/2011 for the course PHY 2048 taught by Professor George during the Fall '10 term at Edison State College.

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Lesson_Text_1.3 - 1 Lesson 1.3 Vectors 1. Vector Addition...

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