Understanding Vector Magnitude Calculation

The problem asks us to find the magnitude of vector \(\overrightarrow b\), given the magnitudes of vector \(\overrightarrow a\), the difference between \(\overrightarrow a\) and \(\overrightarrow b\) (which involves vector subtraction), and the sum of \(\overrightarrow a\) and \(\overrightarrow b\) (which involves vector addition). This is a classic problem in vector algebra requiring a specific formula for vector magnitude calculation.

We are given the following information:

• Magnitude of vector \(\overrightarrow a\), \(\left| {\overrightarrow a } \right| = 5\)
• Magnitude of the difference vector \(\overrightarrow a - \overrightarrow b\), \(\left| {\overrightarrow a - \overrightarrow b } \right| = 8\)
• Magnitude of the sum vector \(\overrightarrow a + \overrightarrow b\), \(\left| {\overrightarrow a + \overrightarrow b } \right| = 10\)

We need to find the magnitude of vector \(\overrightarrow b\), which is \(\left| {\overrightarrow b } \right|\).

Applying the Parallelogram Law for Vector Magnitudes

To solve this vector magnitude calculation problem, we can use the parallelogram law of vector addition and subtraction. The parallelogram law provides a relationship between the magnitudes of two vectors, the magnitude of their sum (resulting from vector addition), and the magnitude of their difference (resulting from vector subtraction).

The parallelogram law states that for any two vectors \(\overrightarrow a\) and \(\overrightarrow b\):

\[{\left| {\overrightarrow a + \overrightarrow b } \right|^2 + \left| {\overrightarrow a - \overrightarrow b } \right|^2 = 2\left( {{{\left| {\overrightarrow a } \right|}^2} + {{\left| {\overrightarrow b } \right|}^2}} \right)}\]

This formula is fundamental for this type of vector magnitude calculation.

Step-by-Step Vector Magnitude Calculation

Now, let's substitute the given values into the parallelogram law formula to find \(\left| {\overrightarrow b } \right|\).

We have:

• \(\left| {\overrightarrow a } \right| = 5 \implies {\left| {\overrightarrow a } \right|^2} = {5^2} = 25\)
• \(\left| {\overrightarrow a - \overrightarrow b } \right| = 8 \implies {\left| {\overrightarrow a - \overrightarrow b } \right|^2} = {8^2} = 64\)
• \(\left| {\overrightarrow a + \overrightarrow b } \right| = 10 \implies {\left| {\overrightarrow a + \overrightarrow b } \right|^2} = {10^2} = 100\)

Substitute these values into the parallelogram law equation:

\[{100 + 64 = 2\left( {25 + {{\left| {\overrightarrow b } \right|}^2}} \right)}\]

Simplify the equation:

\[{164 = 2\left( {25 + {{\left| {\overrightarrow b } \right|}^2}} \right)}\]

Divide both sides by 2:

\[{\frac{{164}}{2} = 25 + {{\left| {\overrightarrow b } \right|}^2}}\]

\[{82 = 25 + {{\left| {\overrightarrow b } \right|}^2}}\]

Isolate \({\left| {\overrightarrow b } \right|^2}\):

\[{{{\left| {\overrightarrow b } \right|}^2} = 82 - 25}\]

\[{{{\left| {\overrightarrow b } \right|}^2} = 57}\]

To find \(\left| {\overrightarrow b } \right|\), take the square root of both sides:

\[{\left| {\overrightarrow b } \right| = \sqrt{57}}\]

Since magnitude is a non-negative value, we take the positive square root.

Therefore, the result of this vector magnitude calculation is \(\sqrt{57}\).