Show
This set of Physics Multiple Choice Questions & Answers (MCQs) focuses on “Vector Addition – Analytical Method”. 1. On adding two vectors we get _____ a) A vector b) A scalar c) A number d) An operation View Answer Answer: a Explanation: Addition of two vectors gives a vector as the result. The same goes for subtraction and cross multiplication. It is only when two vectors are operated through dot product, we get a scalar as the result. 2. Adding 2î + 7ĵ and î + ĵ gives ______ a) 3î + 8ĵ b) î + 35ĵ c) î + 8ĵ d) 2î + 7ĵ View Answer Answer: a Explanation: When 2î + 7ĵ is added to î + ĵ, the corresponding components get added. Hence, the answer is 3î + 8ĵ. 3. Subtracting 2î + 7ĵ from î + ĵ gives ______ a) -î – 6ĵ b) 3î + 8ĵ c) î + 6ĵ d) 7ĵ View Answer Answer: a Explanation: When 2î + 7ĵ is subtracted from î + ĵ, the corresponding components get subtracted. Hence, the answer is -î – 6ĵ. 4. Adding î + 77ĵ and 7î + ĵ gives ______ a) 8î + 78ĵ b) 0î + 76ĵ c) î + 74ĵ d) 78î + 8ĵ View Answer Answer: a Explanation: When î + 77ĵ is added to 7î + ĵ, the corresponding components get added. Hence, the answer is 8î + 78ĵ. 5. When two vectors in the same direction are added, the magnitude of resulting vector is equal to _______ a) Sum of magnitudes of the vectors b) Difference of magnitudes of the vectors c) Product of magnitudes of the vectors d) Sum of the roots of magnitudes of the vectors View Answer Answer: a Explanation: Consider the graphical representation of these two vectors. When one vector is added to the other in the same direction, the lengths will be added. The resultant vector will bear the resultant length. Length is the magnitude of the vector. Hence the magnitudes add to give the magnitude of the resultant vector.
Check this: Class 11 - Mathematics MCQs | Class 12 - Physics MCQs 6. A vector, 7 units from the origin, along the X axis, is added to vector 11 units from the origin along the Y axis. What is the resultant vector? a) 3î + 8ĵ b) 7î + 11ĵ c) 11î + 7ĵ d) 2î + 7ĵ View Answer Answer: b Explanation: The vector 7 units from the origin and along X axis is 7î. The vector 11 units from the origin and along Y axis is 11ĵ. Hence the sum is 7î + 11ĵ. 7. Unit vector along the vector 4î + 3ĵ is _____ a) (4î + 3ĵ)/5 b) 4î + 3ĵ c) (4î + 3ĵ)/6 d) (4î + 3ĵ)/10 View Answer Answer: a Explanation: Unit vector along 4î + 3ĵ , is obtained by dividing the present vector by its magnitude. The magnitude of the given vector is 5. Hence, the required unit vector is (4î + 3ĵ)/5. 8. Unit vector which is perpendicular to the vector 4î + 3ĵ is _____ a) ĵ b) î c) 4î + 3ĵ d) \(\hat{z}\) View Answer Answer: d Explanation: The dot product of any two vectors which are perpendicular to each other is 0. The dot product of all the vectors in the options with 4î + 3ĵ is non-zero except for \(\hat{z}\). Hence \(\hat{z}\) is perpendicular to the given vector. 9. The vector 40î + 30ĵ is added to a vector. The result gives 15î + 3ĵ as the answer. The unknown vector is _____ a) -25î – 27ĵ b) 25î + 27ĵ c) -25î + 27ĵ d) 25î – 27ĵ View Answer Answer: a Explanation: The required vector can be obtained by subtracting 40î + 30ĵ from 15î + 3ĵ. The result gives -25î – 27ĵ. One should pay attention on which vector has to be subtracted from which one. 10. Calculating the relative velocity is an example of ______ a) Vector addition b) Vector subtraction c) Vector multiplication d) Vector division View Answer Answer: b 11. A vector, 5 units from the origin, along the X axis, is added to vector 2 units from the origin along the Y axis. What is the resultant vector? a) 3î + 8ĵ b) 5î + 2ĵ c) 2î + 5ĵ d) 2î + 7ĵ View Answer Answer: b Explanation: The vector 5 units from the origin and along X axis is 5î. The vector 2 units from the origin and along Y axis is 2ĵ. Hence the sum is 5î + 2ĵ. 12. A vector, 14 units from the origin, along the X axis, is added to vector 16 units from the origin along the Z axis. What is the resultant vector? a) 3î + 8\(\hat{z}\) b) 14î + 16\(\hat{z}\) c) 16î + 14\(\hat{z}\) d) 2î + 7\(\hat{z}\) View Answer Answer: b Explanation: The vector 14 units from the origin and along X axis is 14î. The vector 16 units from the origin and along Y axis is 16\(\hat{z}\). Hence the sum is 14î + 16\(\hat{z}\). Sanfoundry Global Education & Learning Series – Physics – Class 11. To practice all areas of Physics, here is complete set of 1000+ Multiple Choice Questions and Answers. Next Steps:
 Subscribe to his free Masterclasses at Youtube & technical discussions at Telegram SanfoundryClasses.
In vector geometry, the resultant vector is defined as: “A resultant vector is a combination or, in simpler words, can be defined as the sum of two or more vectors which has its own magnitude and direction.” In this topic, we will be covering the following concepts:
What Is A Resultant Vector?A resultant vector is a vector that gives the combined effect of all the vectors. When we add two or more vectors, the outcome is the resultant vector. Let’s explore this concept with a simple, practical example. Suppose there is a beam with two boxes lying on it, as shown in the figure below: Will you be able to calculate the weight of the beam and the weight of the two boxes? Yes! You can, as you are going to be familiar with the concept of the resultant vector. In this case, the resultant vector will be the sum of the forces acting on the two boxes, i.e., the boxes’ weight, which will be equal and opposite to the weight of the beam. In this case, the resultant vector will be the sum of two forces as both are parallel and pointing in the same direction. Suppose there are three vectors in a plane, vector A, B and C. There resultant R can be calculated by adding all three vectors. The resultant R can be determined accurately by drawing a properly scaled and accurate vector addition diagram is shown in the figure below: A+B+C = R Let us have a better understanding of the concept with the help of an example. Example 1 Calculate the resultant vector of three parallel forces pointing upwards. OA = 5N, OB = 10N and OC = 15N. Solution As we know that the resultant vector is given as: R = OA + OB +OC R = 5 + 10 + 15 R = 30N Example 2 Find out the resultant vector of the given vectors OA= (3,4) and OB= (5,7). Solution Adding the x-components to find Rx and y-components to calculate RY. RX=3+5 RX =8 Ry=4+7 Ry =11 So, the resultant vector is R=(8,11) How To Find The Resultant VectorsVectors can be added geometrically by drawing them using a common scale according to the head-to-tail convention, which is defined as “Join the tail of the first vector with the head of the second vector, which will give another vector whose head is joined with the head of the second vector and tail of the first vector…” …this is called a resultant vector. Steps To Find Out The Resultant Vector Using Head-To-Tail RuleFollowing are the steps to be followed to add two vectors and find out the resultant vector:
Example 3 Consider a ship sailing at 45o north-east. Then it changes its course in a direction 165o towards the north. Draw the resultant vector. Solution Resultant Vector Of More Than Two VectorsThe rules for finding the resultant of a vector or adding more than two vectors can be protracted to any number of vectors. R=A+B+C+…………………………. Suppose there are three A, B, and C vectors, as shown in the figures below. To add these vectors, draw them according to the head-to-tail rule such that the head of one vector coincides with the other vector. So, the resultant vector is given as follow: R=A+B+C Note: Vector addition is commutative in nature; the sum is independent of the order of addition. R=A+B+C=C+B+C Calculating Resultant Vector Using Rectangular ComponentsFinding a resultant vector using components of a vector is known as an analytical method; this method is more mathematical than geometrical and can be regarded as more accurate and precise than the geometrical method, i.e., configuring using the head-to-tail rule. Suppose there are two vectors A and B, making angles θA and θB respectively with the positive x-axis. These vectors will be resolved into their components. They will be used to calculate the resultant x and y components of the resultant vector R, which will be the sum of the two vectors’ x and y components separately. R = A+B RX = AX + BX eq 1 RY = AY + BY eq 2 Since, by rectangular components R = RX + RX eq 3 Now, putting the values of eq 1 and eq 2 in eq 3 R = (AX + BX) + (AY + BY) By rectangular component, the magnitude of the resultant vector is given as |R| = √ ((Rx )2+( Ry)2) |R| = √ ((Ax + BX )2+( Ay + BY)2) By rectangular components direction of the resultant vector is defined as: θ = tan-1 (RY / Rx) The same method will be applicable for any number of vectors A, B, C, D…… to find out the resultant vector R. R = A+B+C+…… RX = AX+BX+CX+….. RY = AY+BY+CY+…… R = RX + RX θ = tan-1 (RY / Rx) Finding Resultant Vector Using Parallelogram MethodAccording to the law of parallelogram vector addition: “If two vectors acting, at once, at a point can be represented by the adjacent sides of a parallelogram drawn from a point, then the resultant vector is represented by the diagonal of the parallelogram passing through that point.” Consider two vectors A and B acting at a point and represented by the two sides of a parallelogram as shown in the figure. θ is the angle between vectors A and B, and R is said to be the resultant vector. Then, according to the parallelogram law of vector addition, the diagonal of the parallelogram represents the resultant of vectors A and B. Mathematical DerivationBelow given is the mathematical derivation: R=A+B Now, expand S to T and draw QT perpendicular to OT. From triangle OTQ, SQ2=OT2+TQ2 eq 1.4 SQ2=(OS+ST)2+TQ2 In triangle STQ, cosθ=ST/SQ SQcosθ=ST Also, sinθ=TQ/SQ TQ=SQsinθ Putting in eq 1.4 gives, |SQ|=√((A+SQsinθ)2+(SQcosθ)2) Let, SQ=OP=D |SQ||=√((A+Dsinθ)2+(Dcosθ)2) Solving the above equation gives, |SQ|= √(A2+2ADcosθ+D2) So, |SQ| gives the magnitude of the resultant vector. Now finding out the direction of the resultant vector, tanφ = TQ/SQ φ = tan-1 (TQ/OT) tanφ = TQ/ (OS+ST) tanφ = Dsinθ/A+Dcosθ φ = tan –1(Dsinθ/A+Dcosθ) Let’s have a better understanding with the help of an example. Example 4 A force of 12N is making an angle of 45o with the positive x-axis, and the second force of 24N is making an angle of 120o with the positive x-axis. Calculate the magnitude of the resultant force. Solution By resolving the vector into its rectangular components, we know that RX = F1X+F2X RY= F1Y+F2Y |R| = √ ((Rx )2+( Ry)2) eq 1.1 Calculating the values of |RX| and |RY|, |Rx| = |F1X| + |F2X| eq 1.2 |F1X |=F1cosθ1 |F1X |=12cos45 |F1X |=8.48N |F2X |=F2cosθ2 |F2X |=24cos120 |F2x|= -12N Putting the values in eq 1.2 gives, |Rx| = 8.48+(-12) |Rx| = -3.52N Now, finding the y-component of the resultant vector |RY| = |F1Y| + |F2Y| eq 1.3 |F1Y |=F1sinθ1 |F1Y |=12sin45 |F1Y|=8.48N |F2Y |=F2 sinθ2 |F2Y |=24sin120 |F2Y |= 20.78N Putting the values in eq 1.2 gives, |Ry | = 8.48+20.78 |Ry | = 29.26N Now, putting the values in eq 1.1 to calculate the magnitude of the resultant vector R, |R| = √ ((-3.52 )2+( 29.26)2) |R| = √ (12.4+856.14) |R| = 29.5 N So, the magnitude of the resultant vector R is 29.5N. Example 5 Two forces of magnitude 5N and 10N are inclined at an angle of 30o. Calculate the magnitude and direction of the resultant vector using parallelogram law. Solution Given that there are two forces F 1 = 5N and F 2 = 10N and angle θ=30o. Using formula, |R|= √(F12+2F1F2cosθ+F22) |R|= √ ((5)2+2(5)(10) cos30+(10)2) |R| =14.54N φ = tan –1(F2sinθ/F1+F2cosθ) φ = tan-1 (10sin30/(5+10cos30)) φ = 20.1o So, the magnitude of the resultant vector R is 14.54N, and the direction is 20.1o. Practice Problems
All the vector diagrams are constructed by using GeoGebra. Previous Lesson | Main Page | Next Lesson |
When two vectors in the same direction are added, the magnitude of resultant vector is equal to
Pos Terkait
Copyright © 2024 membukakan Inc.
