Solving Vector Equations Magnitude |a+b| + |a-b|

Introduction to Vector Equations and Magnitude

In the realm of mathematics, particularly in vector algebra, understanding how to manipulate and solve vector equations is crucial. Vector equations often involve finding the magnitude of vectors, which represents their length or size. The magnitude of a vector is a scalar quantity and is always non-negative. This article delves into solving a specific vector equation involving the sum and difference of two vectors, a and b. To effectively solve such equations, it's essential to grasp the concepts of vector addition, vector subtraction, and the formula for calculating the magnitude of a vector.

Vector addition is performed component-wise. If we have two vectors, ${ a = a_xi + a_yj }$ and ${ b = b_xi + b_yj }$, their sum, ${ a + b }$, is given by ${ (a_x + b_x)i + (a_y + b_y)j }$. Similarly, vector subtraction is also performed component-wise, with the difference, ${ a - b }$, calculated as ${ (a_x - b_x)i + (a_y - b_y)j }$. The magnitude of a vector, denoted as ${ |v| }$, where ${ v = xi + yj }$, is found using the Pythagorean theorem: ${ |v| = \sqrt{x^2 + y^2} }$. This formula calculates the length of the vector from the origin to the point ${ (x, y) }$ in the Cartesian plane. Understanding these basic operations and concepts is fundamental to tackling more complex vector problems. We will use these principles to solve the given vector equation and determine the correct answer from the provided options.

Problem Statement: |a+b| + |a-b| Where a = 6i + 3j and b = i - 2j

We are tasked with solving the vector equation ${ |a + b| + |a - b| }$, given the vectors ${ a = 6i + 3j }$ and ${ b = i - 2j }$. This problem requires us to first perform vector addition and vector subtraction, and then calculate the magnitude of the resulting vectors. To begin, let's find ${ a + b }$. Adding the corresponding components of a and b, we get:

${ a + b = (6i + 3j) + (i - 2j) = (6 + 1)i + (3 - 2)j = 7i + j }$

Next, we find ${ a - b }$ by subtracting the components of b from a:

${ a - b = (6i + 3j) - (i - 2j) = (6 - 1)i + (3 + 2)j = 5i + 5j }$

Now that we have ${ a + b }$ and ${ a - b }$, we need to calculate their magnitudes. The magnitude of ${ a + b }$, denoted as ${ |a + b| }$, is calculated using the formula ${ |v| = \sqrt{x^2 + y^2} }$. For ${ a + b = 7i + j }$, this gives us:

${ |a + b| = \sqrt{7^2 + 1^2} = \sqrt{49 + 1} = \sqrt{50} }$

Similarly, the magnitude of ${ a - b }$, denoted as ${ |a - b| }$, is calculated for ${ a - b = 5i + 5j }$:

${ |a - b| = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} }$

Finally, we add the magnitudes ${ |a + b| }$ and ${ |a - b| }$ to solve the vector equation:

${ |a + b| + |a - b| = \sqrt{50} + \sqrt{50} = 2\sqrt{50} }$

We can simplify ${ \sqrt{50} }$ as ${ \sqrt{25 \times 2} = 5\sqrt{2} }$, so:

${ 2\sqrt{50} = 2(5\sqrt{2}) = 10\sqrt{2} }$

However, this result does not directly match any of the given options. Let's re-evaluate our calculations and the given options to ensure accuracy.

Detailed Solution Steps and Calculations

Let's meticulously re-examine our solution steps to ensure no errors were made in the calculations. The problem requires us to find the value of ${ |a + b| + |a - b| }$, where ${ a = 6i + 3j }$ and ${ b = i - 2j }$. We will retrace our steps, verifying each calculation to confirm our answer.

First, we calculate ${ a + b }$. Adding the corresponding components of a and b, we have:

${ a + b = (6i + 3j) + (i - 2j) = (6 + 1)i + (3 - 2)j = 7i + j }$

This step appears correct. Next, we calculate ${ a - b }$:

${ a - b = (6i + 3j) - (i - 2j) = (6 - 1)i + (3 - (-2))j = 5i + 5j }$

This step also seems accurate. Now, we calculate the magnitude of ${ a + b }$, which is ${ |a + b| }$. Using the formula ${ |v| = \sqrt{x^2 + y^2} }$, we get:

${ |a + b| = \sqrt{7^2 + 1^2} = \sqrt{49 + 1} = \sqrt{50} }$

The calculation for ${ |a + b| }$ is correct. We then calculate the magnitude of ${ a - b }$, which is ${ |a - b| }$:

${ |a - b| = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} }$

This calculation is also correct. Finally, we add the magnitudes ${ |a + b| }$ and ${ |a - b| }$:

${ |a + b| + |a - b| = \sqrt{50} + \sqrt{50} = 2\sqrt{50} }$

We simplify ${ \sqrt{50} }$ as ${ \sqrt{25 \times 2} = 5\sqrt{2} }$, so:

${ 2\sqrt{50} = 2(5\sqrt{2}) = 10\sqrt{2} }$

Our result is still ${ 10\sqrt{2} }$, which does not match any of the provided options (A. ${ 2\sqrt{5} }$, B. ${ -2\sqrt{5} }$, C. 0, D. ${ 4\sqrt{3} }$ or ${ 4\sqrt{5} }$). It seems we need to revisit our simplification or look for an equivalent form that matches one of the options. Let's examine the options more closely.

We have ${ 2\sqrt{50} }$. We can rewrite this as:

${ 2\sqrt{50} = 2\sqrt{25 \times 2} = 2 \times 5\sqrt{2} = 10\sqrt{2} }$

However, we can also express ${ \sqrt{50} }$ as ${ \sqrt{2 \times 25} = 5\sqrt{2} }$. So, ${ 2\sqrt{50} = 2 \times 5\sqrt{2} = 10\sqrt{2} }$.

Let’s try another approach to see if we can match the options. If we look at option D, ${ 4\sqrt{5} }$, let's see if we can manipulate our result to match this form. We have ${ 10\sqrt{2} }$, and ${ 4\sqrt{5} }$ is approximately equal to ${ \sqrt{16 \times 5} = \sqrt{80} }$. Our result, ${ 10\sqrt{2} }$, is equal to ${ \sqrt{100 \times 2} = \sqrt{200} }$. These are not equal.

Upon further review, we realize a potential simplification error. Let's go back to ${ 2\sqrt{50} }$. We can rewrite ${ \sqrt{50} }$ as ${ \sqrt{25 imes 2} = 5\sqrt{2} }$. Thus,

${ 2\sqrt{50} = 2 imes 5\sqrt{2} = 10\sqrt{2} }$

This still doesn't match any of the options. However, let's consider if there was a mistake in copying the options or the question itself. The closest option we can get to by approximation is D, ${ 4\sqrt{5} }$, which is approximately ${ 4 imes 2.236 = 8.944 }$. Our answer, ${ 10\sqrt{2} }$, is approximately ${ 10 imes 1.414 = 14.14 }$. There is a significant difference.

Given the calculations and the options, it's possible there might be an error in the provided options or the original question. Based on our calculations, the correct answer should be ${ 10\sqrt{2} }$, which is not listed among the options. However, if we consider a typo in the options, it's plausible that option D, ${ 4\sqrt{5} }$, was intended to be ${ 10\sqrt{2} }$ or a close approximation.

Conclusion and Final Answer

After a thorough step-by-step solution and verification, we found that the value of ${ |a + b| + |a - b| }$ for ${ a = 6i + 3j }$ and ${ b = i - 2j }$ is ${ 10\sqrt{2} }$. This result was obtained by first calculating the vectors ${ a + b }$ and ${ a - b }$, then finding their magnitudes, and finally summing the magnitudes. Our detailed calculations are as follows:

1. Calculate ${ a + b = (6i + 3j) + (i - 2j) = 7i + j }$.
2. Calculate ${ a - b = (6i + 3j) - (i - 2j) = 5i + 5j }$.
3. Calculate ${ |a + b| = \sqrt{7^2 + 1^2} = \sqrt{50} }$.
4. Calculate ${ |a - b| = \sqrt{5^2 + 5^2} = \sqrt{50} }$.
5. Sum the magnitudes: ${ |a + b| + |a - b| = \sqrt{50} + \sqrt{50} = 2\sqrt{50} = 10\sqrt{2} }$.

However, the correct answer, ${ 10\sqrt{2} }$, does not match any of the provided options (A. ${ 2\sqrt{5} }$, B. ${ -2\sqrt{5} }$, C. 0, D. ${ 4\sqrt{3} }$ or ${ 4\sqrt{5} }$). This discrepancy suggests a possible error in the options or the question itself.

Based on our rigorous calculations, we can confidently state that the correct answer is ${ 10\sqrt{2} }$. If we had to choose the closest option, it would be option D, ${ 4\sqrt{5} }$, but it is significantly different from our calculated result. Therefore, we conclude that there may be an issue with the provided options.

Final Answer: ${ 10\sqrt{2} }$