Solving |a+b|+|a-b| Vector Equation With A = 6i + 3j And B = I - 2j

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Introduction

In the realm of vector algebra, understanding vector operations and their magnitudes is crucial. This article delves into solving a specific vector equation involving the magnitudes of vector sums and differences. We aim to provide a comprehensive, step-by-step solution to the problem, making it accessible to students and enthusiasts alike. Our focus will be on clarity and thoroughness, ensuring that each step is well-explained and easy to follow. The problem at hand involves finding the value of the expression |a+b|+|a-b|, where a and b are given vectors. This type of problem often appears in introductory linear algebra courses and serves as an excellent exercise in applying fundamental vector concepts. Understanding how to solve such equations is essential for more advanced topics in physics and engineering, where vectors are used extensively to represent physical quantities. By mastering these foundational skills, you'll be well-equipped to tackle more complex problems in vector analysis. We will begin by defining the vectors a and b, then compute their sum and difference. Following this, we will calculate the magnitudes of the resulting vectors and, finally, add them together to arrive at the solution. This structured approach will not only help in solving this particular problem but also in developing a systematic approach to solving other vector equations. Throughout this article, we will emphasize the geometric interpretation of vector addition and subtraction, as well as the concept of magnitude. Visualizing these operations can greatly enhance understanding and make problem-solving more intuitive. We will also highlight common pitfalls and mistakes to avoid, ensuring a thorough grasp of the concepts. So, let's embark on this journey to unravel the solution to the given vector equation, enhancing our understanding of vector algebra along the way.

Problem Statement

The problem we are tackling is to find the value of the expression |a+b|+|a-b|, given the vectors a = 6i + 3j and b = i - 2j. This problem tests our understanding of vector addition, vector subtraction, and the calculation of vector magnitudes. The vectors a and b are expressed in component form, where i and j are the unit vectors along the x and y axes, respectively. This representation is common in many applications, as it allows for easy algebraic manipulation of vectors. The magnitude of a vector, denoted by | |, represents its length and is a scalar quantity. Calculating the magnitude involves using the Pythagorean theorem, which is a fundamental concept in geometry and vector algebra. To solve this problem, we will first compute the vector sum a+b and the vector difference a-b. Vector addition and subtraction are performed component-wise, meaning we add or subtract the corresponding components of the vectors. This process is straightforward but requires careful attention to signs and coefficients. After finding the sum and difference vectors, we will calculate their magnitudes. The magnitude of a vector v = xi + yj is given by |v| = √(x² + y²). This formula is a direct application of the Pythagorean theorem in two dimensions. Once we have the magnitudes of a+b and a-b, we simply add them together to obtain the final answer. This step is a simple arithmetic operation but is crucial for completing the problem correctly. The problem's solution is one of the multiple-choice options provided, making it a typical exam-style question. However, the focus here is not just on finding the correct answer but also on understanding the underlying concepts and the steps involved in solving the problem. By carefully working through the solution, we can reinforce our knowledge of vector algebra and improve our problem-solving skills.

Step 1: Calculate a + b

To begin solving the vector equation, the first crucial step is to calculate the vector sum a + b. Given the vectors a = 6i + 3j and b = i - 2j, we perform vector addition by adding the corresponding components. This process involves adding the i components together and the j components together. The i component of a is 6, and the i component of b is 1. Adding these together gives us 6 + 1 = 7. So, the i component of the resulting vector a + b is 7. Similarly, the j component of a is 3, and the j component of b is -2. Adding these together gives us 3 + (-2) = 1. Thus, the j component of the resulting vector a + b is 1. Therefore, the vector sum a + b is 7i + 1j, which can be simply written as 7i + j. This resulting vector represents the sum of the two original vectors in the same component form. It's important to remember that vector addition is commutative, meaning that a + b is the same as b + a. This property can be helpful in verifying our calculations. Visualizing vector addition can also aid in understanding the process. Geometrically, adding vectors involves placing the tail of one vector at the head of the other, and the resultant vector is the vector drawn from the tail of the first vector to the head of the second. In this case, we can imagine placing the vector b at the end of the vector a, and the resulting vector a + b would extend from the origin to the new endpoint. This geometric interpretation provides a visual check on our algebraic calculations. By accurately calculating a + b, we set the stage for the next steps in solving the problem, which involve finding the magnitude of this vector and then calculating the vector difference a - b.

Step 2: Calculate a - b

Having computed the vector sum a + b, the next essential step in solving the given vector equation is to calculate the vector difference a - b. Similar to vector addition, vector subtraction is also performed component-wise. Given the vectors a = 6i + 3j and b = i - 2j, we subtract the corresponding components of b from a. This means subtracting the i component of b from the i component of a, and similarly for the j components. The i component of a is 6, and the i component of b is 1. Subtracting these gives us 6 - 1 = 5. Thus, the i component of the resulting vector a - b is 5. Next, we subtract the j component of b from the j component of a. The j component of a is 3, and the j component of b is -2. Subtracting these gives us 3 - (-2) = 3 + 2 = 5. Therefore, the j component of the resulting vector a - b is 5. Combining these components, the vector difference a - b is 5i + 5j. It's important to note that vector subtraction is not commutative; that is, a - b is not the same as b - a. In fact, a - b is the negative of b - a. This distinction is crucial in vector algebra and must be kept in mind when performing calculations. Geometrically, vector subtraction can be thought of as adding the negative of the vector being subtracted. In this case, a - b is equivalent to a + (-b). The negative of vector b has the same magnitude as b but points in the opposite direction. By visualizing this, we can gain a better understanding of the subtraction process. Accurately calculating a - b is a critical step in solving the problem, as it sets the stage for finding the magnitude of this vector, which will then be combined with the magnitude of a + b to arrive at the final answer. Understanding both the algebraic and geometric interpretations of vector subtraction is key to mastering vector operations.

Step 3: Calculate |a + b|

Now that we have calculated the vector sum a + b = 7i + j, the next step is to find its magnitude, denoted as |a + b|. The magnitude of a vector represents its length and is a scalar quantity. To calculate the magnitude of a vector in component form, we use the Pythagorean theorem. For a vector v = xi + yj, the magnitude |v| is given by the formula √(x² + y²). In our case, the vector a + b has components x = 7 and y = 1. Plugging these values into the formula, we get: |a + b| = √(7² + 1²). Calculating the squares, we have 7² = 49 and 1² = 1. So, the expression becomes: |a + b| = √(49 + 1). Adding the numbers under the square root, we get: |a + b| = √50. We can simplify the square root by factoring out perfect squares. Since 50 = 25 * 2, we can rewrite the expression as: |a + b| = √(25 * 2). Taking the square root of 25, we get 5, so the simplified magnitude is: |a + b| = 5√2. This value represents the length of the vector a + b. It's important to remember that the magnitude is always a non-negative value. Geometrically, the magnitude corresponds to the length of the vector in the coordinate plane. We can visualize this as the hypotenuse of a right triangle with legs of length 7 and 1, corresponding to the i and j components of the vector. The Pythagorean theorem then confirms our calculation of the magnitude. By accurately calculating |a + b|, we have taken a significant step towards solving the original problem. The next step will be to calculate the magnitude of the vector difference a - b, which we will then combine with this result to find the final answer. Understanding the concept of magnitude and its calculation is fundamental in vector algebra and is crucial for various applications in physics and engineering.

Step 4: Calculate |a - b|

Having determined the vector difference a - b = 5i + 5j, we now need to calculate its magnitude, denoted as |a - b|. Similar to finding the magnitude of a + b, we will use the Pythagorean theorem. The formula for the magnitude of a vector v = xi + yj is |v| = √(x² + y²). In this case, the vector a - b has components x = 5 and y = 5. Substituting these values into the formula, we get: |a - b| = √(5² + 5²). Calculating the squares, we find that 5² = 25. Therefore, the expression becomes: |a - b| = √(25 + 25). Adding the numbers under the square root, we have: |a - b| = √50. Notice that this is the same value we obtained for |a + b|. As before, we can simplify the square root by factoring out perfect squares. Since 50 = 25 * 2, we can rewrite the expression as: |a - b| = √(25 * 2). Taking the square root of 25, we get 5, so the simplified magnitude is: |a - b| = 5√2. This magnitude represents the length of the vector a - b. Geometrically, we can visualize this as the length of the vector extending from the origin to the point (5, 5) in the coordinate plane. This vector forms the hypotenuse of a right triangle with legs of length 5 along both the x and y axes. The Pythagorean theorem confirms our calculation of the magnitude in this geometric context. It's interesting to observe that |a + b| and |a - b| have the same value in this particular problem. This is not always the case, but it highlights the importance of performing the calculations carefully and not making assumptions based on the form of the vectors. With the magnitude of a - b calculated, we are now in the final stage of solving the problem. We have |a + b| = 5√2 and |a - b| = 5√2. The next step is to add these magnitudes together to find the value of the expression |a + b| + |a - b|.

Step 5: Calculate |a + b| + |a - b|

Having computed the magnitudes |a + b| = 5√2 and |a - b| = 5√2, the final step in solving the vector equation is to add these magnitudes together. We are looking for the value of the expression |a + b| + |a - b|. Substituting the values we calculated, we have: |a + b| + |a - b| = 5√2 + 5√2. Since both terms have the same radical part (√2), we can add them like ordinary algebraic terms. This is similar to adding 5x + 5x, which gives 10x. In our case, we have 5√2 + 5√2 = (5 + 5)√2. Adding the coefficients, we get: |a + b| + |a - b| = 10√2. This is the final value of the expression we were asked to find. The result, 10√2, represents the sum of the lengths of the vectors a + b and a - b. It is a scalar quantity, as magnitudes are always non-negative real numbers. Looking back at the multiple-choice options provided, we can see that 10√2 does not directly match any of the given answers (A. 2√5, B. -2√5, C. 0, D. 4√3 or 4√5). However, we need to check if our answer can be expressed in an equivalent form that matches one of the options. Let's examine the options more closely. The correct answer should be 10√2, which can be rewritten as √(10² * 2) = √(100 * 2) = √200. Option D suggests 4√5, which can be written as √(4² * 5) = √(16 * 5) = √80. Another part of Option D is 4√3, which can be written as √(4² * 3) = √(16 * 3) = √48. These don't match √200. Option A is 2√5, which is √(2² * 5) = √(4 * 5) = √20, which also doesn't match. So we have to compare 10√2 with one of them such as 4√5 and square both. We have (10√2)² = 100*2 = 200 and (4√5)² = 16 * 5 = 80, which is different. Hence, there might be an error in the provided options or in our understanding of the options. However, if we closely check, we can rewrite 10√2 as √(100*2) = √200. Now consider √(16*5)=√(80) which is not equal to the answer. So, the answer calculated 10√2 seems correct, and there might be a discrepancy in the options provided. If 4√5 was instead 10√2, that'd be correct. If we had to select from these options, we would re-evaluate our calculations. Let's do it:

  • a+b = 6i+3j + i-2j = 7i+j
  • |a+b| = √(7²+1²) = √50 = 5√2
  • a-b = 6i+3j - (i-2j) = 5i+5j
  • |a-b| = √(5²+5²) = √50 = 5√2
  • |a+b| + |a-b| = 5√2 + 5√2 = 10√2 Thus we can confirm that our calculations are accurate. If one of the options was 10√2, we would select it, but with these available choices, none seems correct. If the question is well-defined, it is possible that the answer is missing from the provided options.

Conclusion

In conclusion, we embarked on a journey to solve the vector equation |a+b|+|a-b|, given the vectors a = 6i + 3j and b = i - 2j. Through a step-by-step approach, we meticulously calculated the vector sum a + b, the vector difference a - b, and their respective magnitudes. We began by computing a + b = 7i + j and a - b = 5i + 5j. These vector operations, performed component-wise, are fundamental to vector algebra. Next, we calculated the magnitude of a + b, which turned out to be |a + b| = √(7² + 1²) = √50 = 5√2. Similarly, we computed the magnitude of a - b, finding |a - b| = √(5² + 5²) = √50 = 5√2. The use of the Pythagorean theorem in these magnitude calculations is a cornerstone of vector analysis, providing a direct link between the components of a vector and its length. Finally, we added the magnitudes together: |a + b| + |a - b| = 5√2 + 5√2 = 10√2. This result represents the sum of the lengths of the vectors a + b and a - b, a scalar quantity that encapsulates the geometric implications of the vector operations. However, upon comparing our result, 10√2, with the provided multiple-choice options (A. 2√5, B. -2√5, C. 0, D. 4√3 or 4√5), we found no direct match. This discrepancy led us to re-evaluate our calculations, ensuring the accuracy of each step. Our thorough review confirmed that the calculations were indeed correct, suggesting a potential issue with the provided options. It is crucial to recognize that, in such cases, the correct approach is to trust the process, verify the calculations, and acknowledge the possibility of an error in the given choices. This exercise underscores the importance of not only understanding vector algebra concepts but also developing critical problem-solving skills, including the ability to identify discrepancies and validate results. While the exact answer may not have been among the options, the process of solving this equation has provided valuable insights into vector operations, magnitude calculations, and the application of the Pythagorean theorem in vector algebra. This knowledge is essential for tackling more complex problems in mathematics, physics, and engineering, where vectors play a crucial role.