The image asks to find the projection of vector $\vec{b}$ onto vector $\vec{a}$, denoted as $\text{Proj}_{\vec{a}} \vec{b}$, where:

$\vec{a} = (-3, 4, -3), \quad \vec{b} = (-3, -1, 1).$

Formula for Vector Projection:

The projection of $\vec{b}$ onto $\vec{a}$ is calculated using the formula:

$\text{Proj}_{\vec{a}} \vec{b} = \frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|^2} \cdot \vec{a}$

Step 1: Calculate the Dot Product $\vec{a} \cdot \vec{b}$

Using the formula for the dot product:

$\vec{a} \cdot \vec{b} = (-3)(-3) + (4)(-1) + (-3)(1)$ $\vec{a} \cdot \vec{b} = 9 - 4 - 3 = 2$

Step 2: Find the Magnitude Squared of $\vec{a}$, $\|\vec{a}\|^2$

The magnitude squared of $\vec{a}$ is:

$\|\vec{a}\|^2 = (-3)^2 + 4^2 + (-3)^2$ $\|\vec{a}\|^2 = 9 + 16 + 9 = 34$

Step 3: Compute the Scalar Coefficient

The scalar coefficient for the projection is:

$\frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|^2} = \frac{2}{34} = \frac{1}{17}$

Step 4: Multiply by $\vec{a}$

Now, multiply $\frac{1}{17}$ by $\vec{a} = (-3, 4, -3)$:

$\text{Proj}_{\vec{a}} \vec{b} = \frac{1}{17} \cdot (-3, 4, -3)$ $\text{Proj}_{\vec{a}} \vec{b} = \left(-\frac{3}{17}, \frac{4}{17}, -\frac{3}{17}\right)$

Final Answer:

$\text{Proj}_{\vec{a}} \vec{b} = \left(-\frac{3}{17}, \frac{4}{17}, -\frac{3}{17}\right)$

Would you like further clarification or detailed steps? Here are five related questions you may explore:

1. How do you calculate the angle between two vectors using the dot product?
2. What is the geometric meaning of a vector projection?
3. How does the formula for projection change for unit vectors?
4. Can the projection formula be extended to higher-dimensional spaces?
5. How does the cross product relate to projections in 3D space?

Tip: Always normalize your direction vectors if you're looking for unit projections.