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

In this article, we will delve into the process of solving a vector equation involving the magnitudes of vector sums and differences. Specifically, we aim to find the value of |a+b| + |a-b|, where the vectors a and b are given as a = 6i + 3j and b = i - 2j. This problem falls under the domain of vector algebra, a crucial area of mathematics and physics. Understanding how to manipulate vectors, compute their magnitudes, and perform operations like addition and subtraction is fundamental to many scientific and engineering applications. Our approach will involve first computing the vectors a+b and a-b, then finding their magnitudes, and finally summing these magnitudes to arrive at the solution. Let's embark on this step-by-step journey to solve this fascinating vector equation.

Step 1: Compute a + b

To begin, we need to determine the vector sum a + b. Vector addition is performed component-wise, meaning we add the corresponding i and j components of the vectors a and b separately. Given a = 6i + 3j and b = i - 2j, we add their i components (6 and 1) and their j components (3 and -2). This straightforward process allows us to combine the vectors into a single resultant vector. The accurate calculation of a + b is critical as it forms the basis for further steps in the problem. Miscalculations here can propagate through the rest of the solution, leading to an incorrect final answer. Thus, careful attention to detail is essential in this initial step. By adding the vectors component-wise, we not only simplify the problem but also adhere to the fundamental principles of vector algebra. This method ensures that the direction and magnitude of the resultant vector are correctly represented, which is crucial in various applications, from physics simulations to engineering designs. Let’s proceed with the calculation and find the vector a + b.

Calculating a + b:

a + b = (6i + 3j) + (i - 2j)

a + b = (6 + 1)i + (3 - 2)j

a + b = 7i + 1j

a + b = 7i + j

Therefore, the vector sum a + b is 7i + j. This result is a new vector, which represents the combined effect of vectors a and b. The i component of the resultant vector is 7, indicating its displacement along the x-axis, and the j component is 1, representing its displacement along the y-axis. This resultant vector is crucial for our next step, where we will calculate its magnitude. Understanding vector addition is paramount in various fields, including physics, engineering, and computer graphics. It allows us to analyze forces, velocities, and displacements in a comprehensive manner. Now that we have a + b, we can proceed to find its magnitude, which will give us a scalar value representing the length of this vector. This value is a critical component in solving the original vector equation.

Step 2: Compute a - b

Next, we need to calculate the vector difference a - b. Similar to vector addition, vector subtraction is also performed component-wise. We subtract the corresponding i and j components of vector b from vector a. Given a = 6i + 3j and b = i - 2j, we subtract the i component of b from the i component of a (6 - 1) and the j component of b from the j component of a (3 - (-2)). This process yields the vector representing the difference between a and b. Accurate computation of a - b is vital because it, like a + b, is a critical component in solving the vector equation. Errors in this step will directly affect the final result. Understanding vector subtraction is essential in physics and engineering, as it helps determine relative velocities, displacements, and forces. For example, in physics, the relative velocity of one object with respect to another is found by subtracting their velocity vectors. In engineering, vector subtraction can be used to calculate the resultant force when multiple forces are acting on an object. By performing the subtraction component-wise, we ensure that both the direction and magnitude of the resultant vector are correctly determined. This meticulous approach is consistent with the principles of vector algebra and allows for precise analysis of vector quantities. Let's proceed with the calculation to find the vector a - b.

Calculating a - b:

a - b = (6i + 3j) - (i - 2j)

a - b = (6 - 1)i + (3 - (-2))j

a - b = 5i + (3 + 2)j

a - b = 5i + 5j

Therefore, the vector difference a - b is 5i + 5j. This vector represents the displacement from the terminal point of vector b to the terminal point of vector a. The components of the vector difference, 5 in the i direction and 5 in the j direction, provide a clear indication of the relative positions of the two vectors. This result is another crucial piece of the puzzle, as we will need to calculate its magnitude in the subsequent steps. Vector subtraction is not only a mathematical operation but also a fundamental concept in physics and engineering. It enables us to analyze the relative motion and forces in a system. Now that we have computed a - b, we can proceed to determine its magnitude, which, along with the magnitude of a + b, will help us solve the original vector equation.

Step 3: Calculate |a + b|

Now that we have computed a + b = 7i + j, we need to find its magnitude, denoted as |a + b|. The magnitude of a vector is its length, which can be calculated using the Pythagorean theorem. For a vector in two dimensions, like ours, the magnitude is the square root of the sum of the squares of its components. In this case, the components of a + b are 7 and 1. Accurately calculating the magnitude is crucial as it converts a vector quantity into a scalar quantity, representing its length. This scalar value is essential for solving the original equation, which involves the sum of magnitudes. The magnitude of a vector is a fundamental concept in physics and engineering. It represents the size or intensity of the vector quantity, such as force, velocity, or displacement. In many applications, we are interested in the magnitude rather than the direction of a vector, making this calculation indispensable. The Pythagorean theorem provides a straightforward method for finding the magnitude of a vector in Euclidean space. By squaring each component, summing them, and then taking the square root, we obtain the length of the vector. This approach is universally applicable to vectors in any number of dimensions, although the complexity of the calculation increases with the number of components. Let’s apply this method to find the magnitude of a + b.

Calculating |a + b|:

|a + b| = √(7² + 1²)

|a + b| = √(49 + 1)

|a + b| = √50

|a + b| = √(25 * 2)

|a + b| = 5√2

Therefore, the magnitude of a + b is 5√2. This scalar value represents the length of the vector 7i + j. It is a positive real number, as magnitudes are always non-negative. This result is a significant step forward in our solution. We have converted the vector a + b into a scalar quantity, which can now be used in the final calculation. The magnitude of a vector provides a measure of its size or intensity, irrespective of its direction. This is often the quantity of interest in many physical and engineering applications. Now that we have |a + b|, we need to calculate |a - b| to complete the problem.

Step 4: Calculate |a - b|

Having computed a - b = 5i + 5j, we now need to determine its magnitude, denoted as |a - b|. As with |a + b|, we use the Pythagorean theorem to calculate the magnitude of a - b. This involves taking the square root of the sum of the squares of its components. The components of a - b are both 5. This calculation will give us the length of the vector a - b, a scalar quantity that is crucial for solving the given vector equation. The magnitude of a vector, as previously discussed, is a fundamental concept representing the length or intensity of the vector. It is a scalar value and is always non-negative. In many physical applications, the magnitude of a vector is more relevant than its direction. For example, when dealing with forces, the magnitude represents the strength of the force, while the direction indicates the line of action. Similarly, in kinematics, the magnitude of the velocity vector is the speed of the object. The Pythagorean theorem provides a reliable method for calculating the magnitude of a vector in any number of dimensions. By squaring the components, summing them, and taking the square root, we obtain the Euclidean length of the vector. This approach is widely used in various fields, including physics, engineering, computer graphics, and data science. Let’s apply this method to calculate the magnitude of a - b.

Calculating |a - b|:

|a - b| = √(5² + 5²)

|a - b| = √(25 + 25)

|a - b| = √50

|a - b| = √(25 * 2)

|a - b| = 5√2

Therefore, the magnitude of a - b is 5√2. This result is another scalar value representing the length of the vector 5i + 5j. We now have the magnitudes of both a + b and a - b, which are essential for the final step of the problem. The equal magnitudes of a + b and a - b in this specific case indicate a certain geometric relationship between the vectors a and b, which could be further explored in a more detailed analysis. However, for the purpose of solving the given equation, we have successfully calculated |a - b|. With both magnitudes known, we can proceed to the final step of adding them together to find the solution.

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

Finally, we can calculate |a + b| + |a - b| by adding the magnitudes we found in the previous steps. We have determined that |a + b| = 5√2 and |a - b| = 5√2. Adding these two values together will give us the final answer to the problem. This step is straightforward but crucial, as it combines the results of all the previous calculations. Accuracy in this final addition is essential to arrive at the correct solution. The sum of the magnitudes |a + b| and |a - b| provides insight into the geometric relationship between the vectors a and b. It represents the sum of the lengths of the vectors resulting from the addition and subtraction of a and b. This type of calculation is fundamental in vector algebra and has applications in various fields, including physics, engineering, and computer graphics. For instance, in physics, it could represent the total displacement or the combined effect of forces acting in different directions. Let’s perform the addition to find the final result.

Calculating |a + b| + |a - b|:

|a + b| + |a - b| = 5√2 + 5√2

|a + b| + |a - b| = 10√2

Therefore, the value of |a + b| + |a - b| is 10√2. This is the final solution to the vector equation. We have successfully computed the magnitudes of a + b and a - b and added them together to obtain this result. The final answer is a scalar value, as it represents the sum of two magnitudes. This completes our step-by-step solution to the problem.

In conclusion, we have successfully solved the vector equation |a + b| + |a - b| for the given vectors a = 6i + 3j and b = i - 2j. Our methodical approach involved first computing the vectors a + b and a - b, then calculating their magnitudes using the Pythagorean theorem, and finally summing these magnitudes to arrive at the solution. The steps included: finding a + b = 7i + j, then calculating |a + b| = 5√2; next, finding a - b = 5i + 5j, and calculating |a - b| = 5√2; and finally, adding the magnitudes to get |a + b| + |a - b| = 10√2. This problem exemplifies the fundamental principles of vector algebra, highlighting the importance of component-wise vector operations and magnitude calculations. Understanding these concepts is crucial for various applications in physics, engineering, and other scientific disciplines. By breaking down the problem into smaller, manageable steps, we were able to systematically solve it and arrive at the correct answer. The final result, 10√2, represents the sum of the lengths of the vectors resulting from the addition and subtraction of a and b. This exercise reinforces the importance of precision and attention to detail in mathematical calculations, especially when dealing with vector quantities. Vector algebra is a powerful tool for analyzing and solving problems involving forces, velocities, and displacements, and mastering its principles is essential for students and professionals in STEM fields.