4. If a = [3, -2, -3] and b = [r, s, t] such that r2 + s2 + t2 = 88, find the maximum value of a . b.

To find the maximum value of the dot product \(\mathbf{a} \cdot \mathbf{b}\) given \(\mathbf{a} = [3, -2, -3]\) and \(\mathbf{b} = [r, s, t]\) with the constraint \(r^2 + s^2 + t^2 = 88\), we can use the Cauchy-Schwarz inequality.

### Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality states that for any two vectors \(\mathbf{u}\) and \(\mathbf{v}\):
\[ |\mathbf{u} \cdot \mathbf{v}| \leq \|\mathbf{u}\| \|\mathbf{v}\| \]

Where \(\|\mathbf{u}\|\) denotes the magnitude (or norm) of vector \(\mathbf{u}\).

### Applying the Inequality

1. Calculate the magnitude of \(\mathbf{a}\):
\[ \|\mathbf{a}\| = \sqrt{3^2 + (-2)^2 + (-3)^2} = \sqrt{9 + 4 + 9} = \sqrt{22} \]

2. Calculate the magnitude of \(\mathbf{b}\):
\[ \|\mathbf{b}\| = \sqrt{r^2 + s^2 + t^2} = \sqrt{88} \]

According to the Cauchy-Schwarz inequality:
\[ |\mathbf{a} \cdot \mathbf{b}| \leq \|\mathbf{a}\| \|\mathbf{b}\| \]

3. Substitute the magnitudes into the inequality:
\[ |\mathbf{a} \cdot \mathbf{b}| \leq \sqrt{22} \cdot \sqrt{88} \]

4. Simplify the expression:
\[ |\mathbf{a} \cdot \mathbf{b}| \leq \sqrt{22 \times 88} \]
\[ |\mathbf{a} \cdot \mathbf{b}| \leq \sqrt{1936} \]
\[ |\mathbf{a} \cdot \mathbf{b}| \leq 44 \]

The maximum value of \(\mathbf{a} \cdot \mathbf{b}\) occurs when \(\mathbf{a}\) and \(\mathbf{b}\) are in the same direction. This means:
\[ \mathbf{b} = k \mathbf{a} \]

For some scalar \(k\), such that:
\[ \sqrt{r^2 + s^2 + t^2} = \sqrt{88} \]

Given:
\[ \mathbf{a} = [3, -2, -3] \]
\[ \mathbf{b} = k[3, -2, -3] \]

Thus:
\[ \mathbf{b} = [3k, -2k, -3k] \]

And:
\[ (3k)^2 + (-2k)^2 + (-3k)^2 = 88 \]
\[ 9k^2 + 4k^2 + 9k^2 = 88 \]
\[ 22k^2 = 88 \]
\[ k^2 = 4 \]
\[ k = \pm 2 \]

For the maximum value, we take the positive value of \(k\):
\[ \mathbf{b} = 2[3, -2, -3] = [6, -4, -6] \]

Then:
\[ \mathbf{a} \cdot \mathbf{b} = [3, -2, -3] \cdot [6, -4, -6] \]
\[ \mathbf{a} \cdot \mathbf{b} = (3 \cdot 6) + (-2 \cdot -4) + (-3 \cdot -6) \]
\[ \mathbf{a} \cdot \mathbf{b} = 18 + 8 + 18 \]
\[ \mathbf{a} \cdot \mathbf{b} = 44 \]

Therefore, the maximum value of \(\mathbf{a} \cdot \mathbf{b}\) is \(44\).

Would you like any further details or have any questions?

Here are some questions you might be interested in:
1. How do you determine the angle between two vectors?
2. What are orthogonal vectors, and how do you identify them?
3. How can you project one vector onto another?
4. What is the geometric interpretation of the cross product?
5. How do you find the area of a parallelogram using vectors?

**Tip:** When maximizing or minimizing dot products subject to constraints, the Cauchy-Schwarz inequality is a powerful tool to use.

Learn how to maximize the dot product of vectors using the Cauchy-Schwarz inequality. Solve the problem involving vectors a = [3, -2, -3] and b = [r, s, t] with the constraint r^2 + s^2 + t^2 = 88. Suitable for advanced high school students and those interested in vector mathematics.

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