UNIT 2. LESSON 3. COMPONENTS OF A VECTOR PDF

Lesson 2.3 Components of Vectors General Physics 11/2 Science, Technology, Engineering, and Mathematics  A force, which is a vector quantity, being experienced by a diagonal beam in the bridge can be resolved into its two components. Determination of these force components is important in c...

Lesson 2.3 Components of Vectors General Physics 11/2 Science, Technology, Engineering, and Mathematics  A force, which is a vector quantity, being experienced by a diagonal beam in the bridge can be resolved into its two components. Determination of these force components is important in calculating and determining its maximum strength. 5  Learning Objectives At the end of the lesson, you should be able to do the following: Apply the analytical method to calculate the components of a vector. Calculate the magnitude and the direction of the resultant vector using its components. 8 Components of a Vector Consider a displacement vector pointing northeast. 10  Components of a Vector It also has a component along the north (vertical axis). 11 Components of a Vector It has a component along the east (horizontal axis). 12 Components of a Vector A vector can be resolved into its components. 13 Components of a Vector Can you resolve the components in this vector? 14  Components of a Vector The direction of the vector can be presented as angle 𝜃 (Greek letter for theta), measured in a counterclockwise direction from the +x-axis. 15  Components of a Vector The sides of the triangle can be defined from the angle. Legend: h = hypotenuse (hyp) o = opposite (opp) a = adjacent (adj) 17  Components of a Vector Trigonometric functions can be expressed as the following equations: 18 Remember In calculating the sine, cosine, and tangent, make sure that your calculator is in degree mode since the angles or directions are given in degrees. Otherwise, you will get a different value for your final answer. 19  Components of a Vector Suppose we have the following displacement vector. 20  Components of a Vector It has a component in the vertical axis. 21  Components of a Vector It has a component in the horizontal axis. 22  Components of a Vector The relationships between its components and its angle can be expressed as the given equations: 23  Components of a Vector Hence, trigonometrically, we can define the components as the following equations. 24 When should you use the given equations for the x- and y- components of a vector? 25  Let’s Practice! What are the x- and y-components of a displacement vector with a magnitude of 50 m and an angle of 30°? 26  Let’s Practice! A car has a displacement of 750 m, 45° north of west. What are the components of the displacement vector? 29  Let’s Practice! A vector located in the second quadrant has an x-component of ‒80 m at an angle of 20°, as shown in the figure below. Find the magnitude of the vector and its y-component. 32  Calculating the Resultant Vector In the same way that a vector can be resolved into its components, a vector’s magnitude and direction can also be calculated using its components. Trigonometric functions involving the angle and the ratio of the sides of a right triangle can also be used to calculate the resultant vector. 35  Calculating the Resultant Vector The resultant vector can be calculated using Pythagorean theorem. 36  Calculating the Resultant Vector The angle or direction of the vector is determined using the inverse tangent function of the ratio of the x- and y- components. To ensure that your final answer is correct it is very important that you check the signs of the x- and y- components. 37 Tips It would be very helpful to draw a sketch to identify the location of the x- and y-components. In this way, you will be able to easily identify the location of the angle (direction) in the Cartesian plane. 38  Let’s Practice! What is the magnitude and direction of a displacement vector if its components are as follows: Ax = 10 m, Ay = 5 m? 39  Check Your Understanding Identify the components of the following vectors. 1. 35 m, 75° 2. 20 km, 10° north of east 3. 65 N, 55° south of west 48  Check Your Understanding Calculate the magnitude and direction of the vector using the following components. 1. Ax = 250 N, Ay = 550 N 2. Ax = 5 km, Ay = 8 km 3. Ax = ‒850 m, Ay = ‒600 m 49  Let’s Sum It Up! A vector at an angle can be resolved into its two components: a component parallel to the horizontal axis (x-axis) and another component parallel to the vertical axis (y-axis). The components of a vector are calculated using the trigonometric functions sine and cosine. 50  Let’s Sum It Up! Angle 𝜃 should be measured in a counterclockwise direction from the positive x- axis to use the given equations for the x- and y- components. The magnitude and direction of a vector can be calculated through its two components using the Pythagorean theorem and the tangent function. 51  Key Formulas Concept Formula Description Use this formula to Components of solve the x- Vectors where component of a Ax is the x- vector. component of the vector A is the magnitude of the vector 𝜃 is the angle measured from the +x-axis 52  Key Formulas Concept Formula Description Use this formula to Components of solve the y- Vectors where component of a Ay is the y- vector. component of the vector A is the magnitude of the vector 𝜃 is the angle measured from the +x-axis 53  Key Formulas Concept Formula Description Use this formula to Components of solve the magnitude Vectors where of a vector. A is the magnitude of the vector Ax is the x- component of the vector Ay is the y- component of the vector 54  Key Formulas Concept Formula Description Use this formula to Components of determine the Vectors where direction of a vector. 𝜃 is the angle Ax is the x- component of the vector Ay is the y- component of the vector 55 Bibliography Bauer, W., and Gary D. Westfall. University Physics with Modern Physics. New York: McGraw-Hill, 2013. Faughn, Jerry S. and Raymond A. Serway. Serway’s College Physics (7th ed). Singapore: Brooks/Cole, 2006. Knight, Randall Dewey. Physics for Scientists and Engineers: a Strategic Approach with Modern Physics. Pearson, 2017. Serway, Raymond A. and John W. Jewett, Jr. Physics for Scientists and Engineers with Modern Physics (9th ed). USA: Brooks/Cole, 2014. Young, Hugh D., Roger A. Freedman, and A. Lewis Ford. Sears and Zemansky’s University Physics with Modern Physics (13th ed). USA: Pearson Education, 2012. 57