General Physics 1 Study Guide PDF

Unit 2: Vectors Lesson 2.6 Operations Using Unit Vectors Contents Introduction 1 Learning Objectives 2 Warm Up 2 Learn about It! 3 Uni...

Unit 2: Vectors Lesson 2.6 Operations Using Unit Vectors Contents Introduction 1 Learning Objectives 2 Warm Up 2 Learn about It! 3 Unit Vector 3 Finding the Unit Vector 4 Vector Addition and Subtraction 5 Multiplying a Vector with a Scalar 7 Vector Multiplication 14 Scalar Product Using Components 14 Vector Product Using Components 21 Key Points 29 Key Formulas 29 Check Your Understanding 32 Challenge Yourself 33 Bibliography 34 Key to Try It! 35 Unit 2: Vectors Lesson 2.6 Operations Using Unit Vectors Introduction Vectors are essential in the development of games, specifically 2D games. Vectors are applied in games to indicate the motion of an object. Numbers are used to indicate the thrust and the gravity that affects an object in situations where jumping is required. Vectors can also be used to specify the movement of an object from its original position. Another application is that vectors can be used for situations that require collisions. These vectors are incorporated in the code as numbers and can be used to manipulate different elements in the game. What are unit vectors? How can these unit vectors be used in mathematical operations? In this lesson, we will understand what unit vectors are and the different mathematical operations that can be done with them. 2.6. Operations Using Unit Vectors 1 Unit 2: Vectors Learning Objectives DepEd Competency In this lesson, you should be able to do the Calculate directions and magnitudes of vector following: (STEM_GP12V-Ia-11). Define a unit vector. Rewrite a vector using its components multiplied by unit vectors. Add and subtract vectors using the vector components. Calculate scalar and vector products using the vector components. Warm Up Vector Pair 10 minutes Unit vectors require the knowledge about vector multiplication. In this activity, you will recall the skills that you have acquired from the past lesson. Materials set of flashcards with answers chalk calculator pen or pencil Procedure 1. Divide the class into two groups. 2. Each group will be given a set of numbers. Each set contains an answer to the following operations using vectors: (a) and ; (b) and ; (c) and ; (d) and. 2.6. Operations Using Unit Vectors 2 Unit 2: Vectors 3. The teacher will write on the board the mathematical operation that the group needs to solve. The values for the vectors are also given. 4. For each mathematical operation, each group will be required to calculate the answer as fast as they can. Look for the answer from the given set of flashcards and post it on the board. 5. Once a group posts their answer, the other group will not be allowed to post their answer anymore. 6. The group who got the highest score after the activity wins. Guide Questions 1. What are the specific equations and concepts you used to solve each problem? 2. What are the difficulties you encountered in performing mathematical operations involving vectors? 3. Can you think of a way to make it easier to perform mathematical operations involving vectors? Explain your answer. Learn about It! There are instances where the separation of magnitude and direction of vectors is convenient to use for basic calculations. This can be done using unit vectors. Unit vectors somehow “normalize” the vector such that the direction is retained but it can be easily scaled up or down by multiplying a scalar value to it. Let us further discuss unit vectors and how they are used in basic mathematical operations. What is a unit vector? Unit Vector A unit vector is a vector that has a magnitude of 1 and has no units. Its main purpose is to specify the direction of a vector. A caret or “hat” (^) is placed above a boldface letter. It is used to differentiate vectors that may or may not have a magnitude of 1. 2.6. Operations Using Unit Vectors 3 Unit 2: Vectors Finding the Unit Vector Any nonzero vector has an equivalent unit vector. It has the same direction as the vector but has a magnitude of 1. Consider vector. Its equivalent unit vector can be determined by dividing the vector with its magnitude, as shown in the expression below: Equation 2.6.1 Vectors are usually written in component form. One way to present these components is through the use of brackets. Components inside the brackets specify the position of the vector in the coordinate system. Let us consider vector with components of. The given components represent the terminal point of the vector. In this example, the vector is in standard position, which means that it started at the origin. The first number in the bracket is the x-component while the second number is the y-component. If there is a third component, another number can be added inside the bracket. Before you can use Equation 2.6.1, you need to calculate the magnitude of the vector first. This is determined using the Pythagorean theorem. From the given components, its magnitude can be calculated as follows. After calculating the magnitude, Equation 2.6.1 can now be used to determine the unit vector. As observed, is now the unit vector of vector. It has a magnitude of 1 but it is parallel or has the same direction as. To check if this is true, the resulting unit vector should have a magnitude of 1, as shown in the calculation as follows. 2.6. Operations Using Unit Vectors 4 Unit 2: Vectors. Aside from components written in brackets, vectors can also be expressed in terms of unit vectors based on their position in a coordinate system. The unit vector is used to indicate a vector in the positive x-direction, while the unit vector is used when the vector points in the y-direction. If there is a third component in the z-axis, the unit vector is used. For example, vector with x- and y-components can be written as. Equation 2.6.2 If there is a component in the z-axis, vector can be written as. Equation 2.6.3 Each term in the two equations above is a vector quantity. How are mathematical operations performed in unit vectors? Vector Addition and Subtraction We learned in the previous lessons that vectors can be added using the graphical and the analytical method. Vectors can also be added and subtracted using their components in terms of the unit vectors. Remember that in the previous lessons, the resultant vector is the sum of the two vectors, in this case, vectors and. These vectors can be expressed 2.6. Operations Using Unit Vectors 5 Unit 2: Vectors in terms of unit vectors as shown below: The resultant vector can be determined by adding and grouping the same terms as shown below: Not all vectors can be found in the x- and y-components. There are instances when there is a third component,. If we say that the two vectors and have three components, their components in terms of unit vectors can be expressed as. If we express the resultant vector in terms of unit vectors, we can use the following expressions: 2.6. Operations Using Unit Vectors 6 Unit 2: Vectors The magnitude of the vector can be determined using the Pythagorean theorem. For example, when the components of vectors and are given, their magnitudes can be determined using the equations below. Multiplying a Vector with a Scalar It is common for a vector to be multiplied by a scalar quantity. Suppose we have a scalar quantity c multiplied to a vector. This will give us. The scalar number can be multiplied to each of the components of. This is shown as follows: A vector given as means that it has twice the magnitude of and points in the same direction as the said vector. A vector that is indicates that it is four times the magnitude of but points in the negative direction, as shown by the negative sign. Remember Always remember the trigonometric functions (sine, cosine, and tangent) and the Pythagorean theorem as you proceed with the lesson. These concepts are essential in all the worked examples to follow. 2.6. Operations Using Unit Vectors 7 Unit 2: Vectors Let’s Practice! Example 1 What is the unit vector of (a) and (b) ? Solution Step 1: Identify what is required in the problem. You are asked to express the vectors in their unit vector form. Step 2: Identify the given in the problem. Both vectors and are given. Step 3: Write the working equation. To calculate the magnitudes of and , we can use the following equations: To determine the unit vectors of both and , we can use the following equations: Step 4: Substitute the given values. For the magnitudes of and ,. 2.6. Operations Using Unit Vectors 8 Unit 2: Vectors For the unit vectors of both and ,. Step 5: Find the answer. The unit vectors of and are and , respectively. Vectors and are equal but are only written in different forms. 1 Try It! What is the unit vector of ? How will you write the unit vector form of in a bracket? 2.6. Operations Using Unit Vectors 9 Unit 2: Vectors Example 2 Three displacement vectors have magnitudes A = 5, B = 10, and C = 20, respectively. Their directions are measured from the +x-axis with angles α = 45°, β = 200°, and 𝛾 = 30°, for vectors , , and , respectively. (a) Find the resultant vector in terms of unit vectors. (b) Find the magnitude and direction of the resultant vector. Solution Step 1: Identify what is required in the problem. You are asked to find the resultant vector in terms of unit vectors as well as its magnitude and direction. Step 2: Identify the given in the problem. The magnitude and direction of the three vectors , , and are given. Step 3: Write the working equation. Before calculating the resultant vector , the components of the vectors should be determined first using the following equations. These equations can be used because the angles are measured from the x-axis. To find the scalar components of the resultant vector , use the following equations: 2.6. Operations Using Unit Vectors 10 Unit 2: Vectors To calculate the magnitude of the resultant vector , Pythagorean theorem can be used. To determine the direction of the resultant vector , the inverse tangent function can be used as. Step 4: Substitute the given values. Determine the components of the vectors. Find the scalar components of the resultant vector. Find the magnitude and direction of the resultant vector. 2.6. Operations Using Unit Vectors 11 Unit 2: Vectors Step 5: Find the answer. The resultant vector is. The magnitude and direction of the resultant vector is 15.29, 41.44°. 2 Try It! Find vector using the given magnitudes and directions of vectors in Example 2. Specify the magnitude and direction of vector. Example 3 Find the magnitude of vector that will satisfy where and. Solution Step 1: Identify what is required in the problem. You are asked to calculate the magnitude of vector. Step 2: Identify the given in the problem. The following equations are given: 2.6. Operations Using Unit Vectors 12 Unit 2: Vectors Step 3: Write the working equation. To determine the magnitude of vector , we need to rearrange the main equation so that we can have the expression for. To calculate the magnitude of vector , the Pythagorean theorem can be used. Step 4: Substitute the given values. Let us now substitute and to the equation below. Regroup and add the same terms accordingly. Determine the magnitude of vector. 2.6. Operations Using Unit Vectors 13 Unit 2: Vectors Step 5: Find the answer. The magnitude of is equal to 7.07. 3 Try It! What is the magnitude of to satisfy the equation ? Consider and ? How can we calculate the scalar and vector products using the components? Vector Multiplication In the previous lesson, we discussed the scalar and vector products using specific equations given the magnitudes of the vectors and the angle between them. However, it can also be calculated if the components of the vectors are given. We will discuss it in detail in the next sections. Scalar Product Using Components From what was discussed previously, a vector can be expressed in terms of the unit vectors , , and. All these unit vectors have magnitudes of 1 and are perpendicular to each other. Let us consider vectors and , and their scalar product. The calculations are shown on the next page. 2.6. Operations Using Unit Vectors 14 Unit 2: Vectors We can rewrite each term from the equation above, such that the unit vectors can be multiplied to each other and be further simplified. To simplify this equation, we have to go back to the equation in determining the scalar product, which is. When vectors are multiplied by themselves, the angle 𝜙 would be equal to 0°. When this angle is substituted to the equation, we will have. Since the unit vectors are perpendicular to each other, a unit vector multiplied to another unit vector will have an angle 𝜙 of 90°. When this angle is substituted to the equation , we will have. 2.6. Operations Using Unit Vectors 15 Unit 2: Vectors Using these relationships and applying it to the expanded version of the scalar product , we will have six terms that will be equal to zero. We are only left with three terms given as. Equation 2.6.4 This means that the scalar product of two vectors is simply the sum of the products of their components. This equation can be used to determine the magnitude and direction of vectors even if they are presented in terms of unit vectors. How can you calculate the scalar product of two vectors? Let’s Practice! Example 4 Find the scalar product of: is 8, 30° and is 10, 110°. Both vectors are measured from the +x-axis. Solution Step 1: Identify what is required in the problem. You are asked to calculate the scalar product. Step 2: Identify the given in the problem. The magnitudes and the directions of both and are given. 2.6. Operations Using Unit Vectors 16 Unit 2: Vectors Step 3: Write the working equation. Before we can use the equations for the scalar product, we need to determine the x- and y-components of both vectors. The following equations can be used since the angles are measured from the +x-axis. The scalar product can be determined using the equation below. Step 4: Substitute the given values. There is no third component for both vectors making the z-component equal to zero, therefore. Step 5: Find the answer. The scalar product of is 13.894. 2.6. Operations Using Unit Vectors 17 Unit 2: Vectors 4 Try It! What is the scalar product if vector A has a value 20 m and found at 100° and vector B has a value 30 m at 150°. Both angles are measured from the +x-axis. Example 5 Find the scalar product if and. Find also the magnitudes of both each vector. Solution Step 1: Identify what is required in the problem. You are asked to calculate the scalar product. Step 2: Identify the given in the problem. Vectors and are both given. Step 3: Write the working equation. To determine the scalar product of , use the following equation: To determine the magnitudes of both and , use the Pythagorean theorem. Step 4: Substitute the given values. For the scalar product : 2.6. Operations Using Unit Vectors 18 Unit 2: Vectors For the magnitudes of vectors and : Step 5: Find the answer. The scalar product is 9.00. The magnitudes vectors A and B are 4.123 and 5.477, respectively. 5 Try It! What are the magnitudes of the following vectors: , , and ? What is the scalar product of vectors A and C? Example 6 Find the angle 𝜙 between vectors: and for. Solution Step 1: Identify what is required in the problem. You are asked to calculate the angle 𝜙 between vectors and. 2.6. Operations Using Unit Vectors 19 Unit 2: Vectors Step 2: Identify the given in the problem. Both vectors and are given. Step 3: Write the working equation. You can derive the equation for 𝜙 using the equations and. So the working equation would be,. To calculate the magnitudes of vectors and , use the Pythagorean theorem. Step 4: Substitute the given values. For the magnitudes of vectors and : For the angle 𝜙 between vectors and , use the inverse cosine function. Step 5: Find the answer. The angle 𝜙 between the two vectors and is 148.38°. 2.6. Operations Using Unit Vectors 20 Unit 2: Vectors 6 Try It! What is the angle 𝜙 between vectors and for ? Vector Product Using Components We can also calculate the vector product using the components in terms of unit vectors. Before we proceed with the calculation of the vector product, let us first check the vector product of the unit vectors. From the previous lesson, we learned that the vector product of any vector with itself is zero. This is also observed when the equation for the magnitude of the scalar product used, where 𝜙 is equal to 90°. Using the said concepts,. In the equation above, we can only conclude that the components of the vector product is zero, while its direction is still undefined. The direction can be determined using the right-hand rule. The following equations result from the right-hand rule, which can be verified using Fig. 2.6.1. Fig. 2.6.1. A right-handed coordinate system 2.6. Operations Using Unit Vectors 21 Unit 2: Vectors Fig. 2.6.1 is a right-handed system. This will be the coordinate system that will be used throughout the lesson. If the left-handed system is used, all the vector products will have the opposite signs compared to the equations above. Let us now proceed to the calculation of the vector product using the components. Consider again that we want to calculate the vector product of. The expansion of the vector product is as shown below: All the terms can be rewritten as follows. We can use the relationship of the vector product of unit vectors shown above to eliminate some of the terms and to simplify some of them. Regrouping the terms will give us. 2.6. Operations Using Unit Vectors 22 Unit 2: Vectors Since we know that the unit vectors , , and correspond to the x-, y-, and z-axes, respectively, we can now say that the components of will be: Equation 2.6.5. These relationships can be applied in the following worked examples. Remember Always keep in mind that the distributive property applies to both scalar and vector products. However, commutative property does not apply to vector product. This means that the order of multiplication matters. It may affect your final answer. Let’s Practice! Example 7 Vector has a magnitude of 20 and lies along the +x-axis. Vector has a magnitude of 10, lies in the xy-plane, and makes 45° with the +x-axis. What is the vector product ? Solution Step 1: Identify what is required in the problem. You are asked to calculate the vector product. Step 2: Identify the given in the problem. Both vectors and are given. 2.6. Operations Using Unit Vectors 23 Unit 2: Vectors Step 3: Write the working equation. Use the following equations to determine the x- and y-components of vectors and. Note that these equations are only used because the angle was measured from the x-axis. Use the equations below to determine the magnitude (C) of the vector product. Step 4: Substitute the given values. Since vector is along the x-axis, the angle would be equal to 0°. Vector is 45° from the x-axis. There is no component along the z-axis. The magnitude of C of the vector product can then be determined. 2.6. Operations Using Unit Vectors 24 Unit 2: Vectors Step 5: Find the answer. The vector product of is. 7 Try It! What is the vector product if lies in the xy-plane, has a magnitude of 35, and makes an angle of 20° from the +x-axis, while vector lies along the +y-axis and has a magnitude of 55? Example 8 Find the vector product if and. Solution Step 1: Identify what is required in the problem. You are asked to calculate the vector product. Step 2: Identify the given in the problem. Vectors and are given. Step 3: Write the working equation. Before you can calculate the vector product, you need to identify the components of the vectors. 2.6. Operations Using Unit Vectors 25 Unit 2: Vectors To calculate the magnitude of the vector product , use the following equations: Step 4: Substitute the given values. Step 5: Find the answer. The vector product is equal to. 8 Try It! What is the vector product if and ? 2.6. Operations Using Unit Vectors 26 Unit 2: Vectors Example 9 Given vectors and , find (a) vector product , and (b) the angle between vectors and. Solution Step 1: Identify what is required in the problem. You are asked to calculate the vector product and the angle between the vectors and. Step 2: Identify the given in the problem. Vectors and are both given. Step 3: Write the working equation. Identify the components of the vectors first. Calculate the magnitude of the vector product. Calculate the magnitudes of A, B, C using the Pythagorean theorem. 2.6. Operations Using Unit Vectors 27 Unit 2: Vectors To determine the angle between the vectors, use the inverse sine function. Step 4: Substitute the given values. The components of the vector product of are: The magnitudes of A, B, and C are: The angle between the vectors is determined by the inverse sine function. Step 5: Find the answer. The vector product is equal to. The angle 𝜙 between the two vectors is 70.58°. 2.6. Operations Using Unit Vectors 28 Unit 2: Vectors 9 Try It! Three vectors , , and are given. Find (a) ; (b) the angle between vectors and ; (c) the angle between and. Key Points ___________________________________________________________________________________________ A unit vector is a vector that has no units but has a magnitude of 1. Its main purpose is to specify the direction of a vector. A caret or “hat” (^) is placed above a boldface letter. Vectors can be added and subtracted if their components are given in terms of unit vectors. The scalar product of two vectors is the sum of the products of their components. The vector product of two vectors can be determined by calculating the scalar components of the cross product vectors. The direction can be specified using the right-handed system. ___________________________________________________________________________________________ Key Formulas ___________________________________________________________________________________________ Concept Formula Description Operations Using Use this formula to Unit Vectors determine the equivalent unit vector of a given vector. where: is the unit vector 2.6. Operations Using Unit Vectors 29 Unit 2: Vectors is the vector is the magnitude of vector Use this formula to write the x- and y-components of where: a vector in terms of unit is the vector vectors. Ax is the x-component of Ay is the y-component of Use this formula to write the x-, y-, and z-components where: of a vector in terms of unit is the vector vectors. Ax is the x-component of Ay is the y-component of Az is the z-component of Use this formula to calculate the scalar product of two where: vectors. is the scalar product of two vectors Ax is the x-component of Bx is the x-component of Ay is the y-component of By is the y-component of 2.6. Operations Using Unit Vectors 30 Unit 2: Vectors Az is the z-component of Bz is the z-component of Use this formula to calculate the components of the magnitude of the vector where product of two vectors. Ax is the x-component of Bx is the x-component of Cx is the x-component of Ay is the y-component of By is the y-component of Cy is the y-component of Az is the z-component of Bz is the z-component of Cz is the z-component of ___________________________________________________________________________________________ 2.6. Operations Using Unit Vectors 31 Unit 2: Vectors Check Your Understanding A. Fill in the missing word(s) to complete each statement. 1. A unit vector has a magnitude of __________. 2. A unit vector specifies the __________ of the vector. 3. A __________ is placed above a boldface character to symbolize the unit vector. 4. The scalar product of two vectors is simply the __________ of their components. 5. The magnitude of vectors can be determined using the __________. B. Write true if the statement is correct. Otherwise, write false. 1. Vector has a magnitude of. 2. The equivalent unit vector of is. 3. These two vectors are the same: and. 4. The left-handed system is used to determine the direction of the vector product of the unit vectors. 5. The unit vectors in the x-, y-, and z-directions are , , and , respectively. C. Solve the following problems. 1. What is the equivalent unit vector of ? 2. Find the equivalent unit vector of. 3. Find the magnitude of the resultant if and. 4. Find the magnitude of the resultant vector if , , and. 2.6. Operations Using Unit Vectors 32 Unit 2: Vectors 5. Find the magnitude of that can satisfy if and. 6. Find the scalar product if and. 7. What is the angle 𝜙 between vectors and for ? 8. What is the magnitude of the vector product if and ? 9. Two vectors and = are given. Find (a) the magnitude of and (b) the angle between vectors and. 10. Given two vectors and. Find (a) and (b) the magnitude of. Challenge Yourself Answer the following questions. 1. After your lesson about unit vectors, your classmate argued that the unit vector of is because the magnitude of vector is 4. Do you agree with your classmate’s argument? Why? Why not? 2. Consider a nonzero vector. How will you write its equivalent unit vector? How about if vector has an angle 𝜃 with respect to the x-axis, what would be its direction? 3. Give a nonzero vector (written in unit vector notation) that would be parallel to. 2.6. Operations Using Unit Vectors 33 Unit 2: Vectors 4. Given two vectors and. (a) Find. (b) Is the magnitude of equal to the magnitude of ? Explain your answer. 5. Consider three nonzero vectors , , and. Is it true that equal to ? Prove your answer. Bibliography Faughn, Jerry S. and Raymond A. Serway. Serway’s College Physics (7th ed). Singapore: Brooks/Cole, 2006. Giancoli, Douglas C. Physics Principles with Applications (7th ed). USA: Pearson Education, 2014. Halliday, David, Robert Resnick and Kenneth Krane. Fundamentals of Physics (5th ed). USA: Wiley, 2002. Knight, Randall D. Physics for Scientists and Engineers: A Strategic Approach (4th ed). USA: Pearson Education, 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. 2.6. Operations Using Unit Vectors 34 Unit 2: Vectors Key to Try It! 1. ; 2. ; E = 79.514; 𝜃 = 30.61° 3. B = 23.22 4. 385.67 5. A= 22.913; B=40.311; C=18.708; = 175 6. 𝜙=72.99° 7. 8. 9. (a) ; (b) ; (c) 2.6. Operations Using Unit Vectors 35