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:

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