Key Concepts and Formulas

• Vector Decomposition: Any vector $\vec{a}$ can be expressed as the sum of its component parallel to a vector $\vec{b}$ ($\vec{a}_{||}$) and its component perpendicular to $\vec{b}$ ($\vec{a}_{\perp}$). Mathematically, $\vec{a} = \vec{a}_{||} + \vec{a}_{\perp}$.
• Vector Magnitude Squared: For a vector $\vec{v} = x\hat{i} + y\hat{j} + z\hat{k}$, its magnitude squared is given by $|\vec{v}|^2 = x^2 + y^2 + z^2$. In this problem, $\alpha^2+\beta^2+\gamma^2 = |\vec{a}|^2$.

Step-by-Step Solution

Step 1: Reconstruct Vector $\vec{a}$ from its Components

• Why this step? We are given the components of $\vec{a}$ along and perpendicular to $\vec{b}$. The fundamental principle of vector decomposition states that the sum of these components equals the original vector. Therefore, we can find $\vec{a}$ by adding the given components.

We are given: $\vec{a}_{||} = \frac{16}{11}(3 \hat{i}+\hat{j}-\hat{k})$ $\vec{a}_{\perp} = \frac{1}{11}(-4 \hat{i}-5 \hat{j}-17 \hat{k})$ Using the decomposition principle, $\vec{a} = \vec{a}_{||} + \vec{a}_{\perp}$: $\vec{a} = \frac{16}{11}(3 \hat{i}+\hat{j}-\hat{k}) + \frac{1}{11}(-4 \hat{i}-5 \hat{j}-17 \hat{k})$ To simplify the addition, we can factor out $\frac{1}{11}$: $\vec{a} = \frac{1}{11} \left[ 16(3 \hat{i}+\hat{j}-\hat{k}) + (-4 \hat{i}-5 \hat{j}-17 \hat{k}) \right]$ Distribute the scalar $16$: $\vec{a} = \frac{1}{11} \left[ (48 \hat{i} + 16 \hat{j} - 16 \hat{k}) + (-4 \hat{i}-5 \hat{j}-17 \hat{k}) \right]$ Combine the coefficients of the like unit vectors: $\vec{a} = \frac{1}{11} \left[ (48 - 4)\hat{i} + (16 - 5)\hat{j} + (-16 - 17)\hat{k} \right]$ Perform the arithmetic: $\vec{a} = \frac{1}{11} \left[ 44\hat{i} + 11\hat{j} - 33\hat{k} \right]$ Distribute the $\frac{1}{11}$: $\vec{a} = 4\hat{i} + \hat{j} - 3\hat{k}$

Step 2: Identify the Scalar Components $\alpha, \beta, \gamma$

• Why this step? The problem states that $\vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}$. By comparing the vector $\vec{a}$ we found in Step 1 with this general form, we can directly determine the values of $\alpha$, $\beta$, and $\gamma$.

From Step 1, we have $\vec{a} = 4\hat{i} + \hat{j} - 3\hat{k}$. Comparing this with $\vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}$, we get: $\alpha = 4$ $\beta = 1$ $\gamma = -3$

Step 3: Calculate $\alpha^2+\beta^2+\gamma^2$

• Why this step? This is the final quantity the problem asks us to find. We use the scalar components identified in Step 2.

Substitute the values of $\alpha, \beta, \gamma$ into the expression: $\alpha^2+\beta^2+\gamma^2 = (4)^2 + (1)^2 + (-3)^2$ Calculate the squares: $= 16 + 1 + 9$ Sum the results: $= 26$

Common Mistakes & Tips

• Component vs. Projection: Ensure you are working with vector components as given, not scalar projections (which are magnitudes). The problem provides vectors.
• Arithmetic Errors: Be meticulous with calculations involving fractions and negative signs, as these are common sources of error.
• Understanding $|\vec{a}|^2$: Recognize that $\alpha^2+\beta^2+\gamma^2$ is simply the square of the magnitude of vector $\vec{a}$.

Summary

The problem involves decomposing a vector into its orthogonal components. By applying the fundamental principle that the sum of these components equals the original vector, we first reconstructed $\vec{a}$. We then identified its scalar components $\alpha, \beta, \gamma$ by comparing with the given form. Finally, we calculated the required expression $\alpha^2+\beta^2+\gamma^2$.

The final answer is \boxed{26}.