Lesson 2.2 Vector Addition Through Graphical Method Vectors in One Dimension • • If two or more vectors are pointing in the same direction, it means that they are parallel to each other, as shown in Fig. 2.2.2. When two vectors have opposite directions but have the same magnitude, they are called antiparallel. *Notice that the length of is similar to the length of , but it is pointing in the opposite direction. Addition of Vectors • • To get the total displacement, we only need to consider the initial and the starting points. The initial point of the total displacement is the starting point of A while the endpoint is the endpoint of vector B. • • We call this displacement as R , also called the vector sum or resultant. It can be How are vectors added graphically? #1 - Suppose a person covered two different displacements: A = 30 m, 25° north of east and B = 55 m, 70° south of east. What is her total displacement? 1. Draw vector on a graphing or bond paper. Make sure that the magnitude is represented by a proper scale. expressed mathematically as R = A + B. Head to tail method, it is where the tail of the second vector is drawn at the head or tip of the first vector. Graphical Method of Adding Vectors 2. Draw vector B using the same scale that you used1 in the previous vector. Its tail should start from the tip of A. Lesson 2.2 Vector Addition Through Graphical Method 3. Draw an arrow connecting the tail of A and the tip of B . This is the resultant vector R . 4. Measure the length of and use the scale to find its real length. Use the protractor to measure the angle. 5. If there is another vector, simply connect it to the tail of the previous vector and follow similar steps. Let's Practice! *Scale: 1cm = 20m *Scale: 1cm = 1 000m Lesson 2.3 Components of Vectors Even though the graphical method of adding vectors is effective in visualizing the process, its final answer is not that accurate due to the precision of the measuring devices used. This is the reason why the analytical method is used to get answers more accurately. Let us first discuss a topic essential in the analytical method of adding vectors— components of vectors. Components of a Vector • • • • • • A vector directed at an angle from the horizontal or vertical axis can be resolved into its components. Consider a displacement vector pointing in the northeast direction. It can be divided into its two components: a component along the east (horizontal axis) and another component along the north (vertical axis), as shown in Fig. 2.3.2; also composed of a component along the south and another component along the west direction. The components can be calculated if the magnitude and the direction of the vector are given. However, it is important to specify first the reference direction. The direction of the vector can be presented as an angle 𝜃 (Greek letter for theta), measured in a counterclockwise direction from the +x-axis. calculated using the trigonometric functions sine, cosine, and tangent. • The x-component of vector A is given below. • The y-component of vector A is given below. *In both equations, A is the magnitude of the vector and 𝜃 is the angle as measured from the positive x-axis. Let's Practice! Let's Practice! Calculating the Resultant Vector • • A vector’s magnitude and direction can also be calculated using its components. The magnitude of vector can be calculated using the equation below based on the Pythagorean theorem. • The angle or direction is calculated using the inverse function of tangent as shown in the equation below. • To ensure that your final answer is correct it is very important that you check the signs of the xand y-components. • After identifying the location, check the sign of the angle 𝜃 you calculated. • A positive sign means that you have to measure in a counterclockwise direction from one of the axes; • While a negative sign indicates that you have to measure the angle in a clockwise direction.