If p(x) and q(x) are arbitrary polynomials of degr

Questions: If p(x) and q(x) are arbitrary polynomials of degree at most 2, then the mapping <p, q> = p(-3) q(-3) + p(0) q(0) + p(3) q(3) defines an inner product in P3. Use this inner product to find <p, q>, p, q, and the angle theta between p(x) and q(x) for p(x) = 4 x^2 + 6 and q(x) = 2 x^2 - 2 x. <p, q> = p = q = theta =

Transcript text: If $p(x)$ and $q(x)$ are arbitrary polynomials of degree at most 2 , then the mapping \[

Solution

Solution Steps

To solve this problem, we need to compute the inner product of two polynomials using the given formula. Then, we calculate the norms of each polynomial using the inner product definition. Finally, we use the dot product and norms to find the angle between the polynomials using the cosine formula.

Inner Product Calculation: Substitute the given polynomials into the inner product formula to find $\langle p, q \rangle$.
Norm Calculation: Compute the norm of each polynomial using $\|p\| = \sqrt{\langle p, p \rangle}$ and $\|q\| = \sqrt{\langle q, q \rangle}$.
Angle Calculation: Use the formula $\cos(\theta) = \frac{\langle p, q \rangle}{\|p\| \|q\|}$ to find the angle $\theta$.

Step 1: Inner Product Calculation

We compute the inner product $\langle p, q \rangle$ using the formula: \[ \langle p, q \rangle = p(-3) q(-3) + p(0) q(0) + p(3) q(3) \] Substituting the values, we find: \[ \langle p, q \rangle = 1512 \]

Step 2: Norm of $p(x)$

The norm of the polynomial $p(x)$ is calculated as: \[ \|p\| = \sqrt{\langle p, p \rangle} \] From our calculations, we have: \[ \|p\| \approx 59.6992 \]

Step 3: Norm of $q(x)$

Similarly, the norm of the polynomial $q(x)$ is given by: \[ \|q\| = \sqrt{\langle q, q \rangle} \] The computed value is: \[ \|q\| \approx 26.8328 \]

Step 4: Angle Between $p(x)$ and $q(x)$

The angle $\theta$ between the two polynomials is found using the formula: \[ \cos(\theta) = \frac{\langle p, q \rangle}{\|p\| \|q\|} \] Calculating $\theta$: \[ \theta \approx 0.3366 \text{ radians} \]

Final Answer

\[ \langle p, q \rangle = 1512, \quad \|p\| \approx 59.6992, \quad \|q\| \approx 26.8328, \quad \theta \approx 0.3366 \] Thus, the final boxed answers are: \[ \boxed{\langle p, q \rangle = 1512} \] \[ \boxed{\|p\| \approx 59.6992} \] \[ \boxed{\|q\| \approx 26.8328} \] \[ \boxed{\theta \approx 0.3366} \]

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