We are given two vectors:

$\vec{a} = 4\hat{i} - 2\hat{j} + 2\hat{k}$ $\vec{b} = 2\hat{i} - 6\hat{j} + 4\hat{k}$

The task is to find the orthogonal projection of vector $\vec{a}$ onto vector $\vec{b}$.

The formula for the projection of vector $\vec{a}$ onto vector $\vec{b}$ is:

$\text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{\vec{b} \cdot \vec{b}} \vec{b}$

Step 1: Compute the dot product $\vec{a} \cdot \vec{b}$

$\vec{a} \cdot \vec{b} = (4)(2) + (-2)(-6) + (2)(4) = 8 + 12 + 8 = 28$

Step 2: Compute the dot product $\vec{b} \cdot \vec{b}$

$\vec{b} \cdot \vec{b} = (2)^2 + (-6)^2 + (4)^2 = 4 + 36 + 16 = 56$

Step 3: Use the projection formula

$\text{proj}_{\vec{b}} \vec{a} = \frac{28}{56} \vec{b} = \frac{1}{2} \vec{b}$

Thus,

$\text{proj}_{\vec{b}} \vec{a} = \frac{1}{2} (2\hat{i} - 6\hat{j} + 4\hat{k}) = \hat{i} - 3\hat{j} + 2\hat{k}$

So, the orthogonal projection of $\vec{a}$ onto $\vec{b}$ is:

$\hat{i} - 3\hat{j} + 2\hat{k}$

Would you like more details on this, or have any questions?

Related Questions:

1. How do we compute the dot product of two vectors?
2. What is the geometric meaning of a vector projection?
3. How can we generalize this formula to projections in higher dimensions?
4. What is the difference between the orthogonal projection and the cross product?
5. How can we find the angle between two vectors using the dot product?

Tip: The magnitude of a vector is useful when normalizing vectors, especially in unit vector calculations!