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The problem asks to find both the projection of vector \(\vec{a}\) onto vector \(\vec{b}\) (\(\text{Proy}_{\vec{b}}\vec{a}\)) and the component of \(\vec{a}\) perpendicular to \(\vec{b}\) (\(\text{Comp}_{\vec{b}^{\perp}}\vec{a}\)).

Given:
- \(\vec{a} = (3, 5)\)
- \(\vec{b} = (6, 1)\)

### Step 1: Find the projection of \(\vec{a}\) onto \(\vec{b}\)
The formula for the projection of \(\vec{a}\) onto \(\vec{b}\) is:
\[
\text{Proy}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{\vec{b} \cdot \vec{b}} \vec{b}
\]

First, calculate the dot products:
1. \(\vec{a} \cdot \vec{b} = 3 \times 6 + 5 \times 1 = 18 + 5 = 23\).
2. \(\vec{b} \cdot \vec{b} = 6 \times 6 + 1 \times 1 = 36 + 1 = 37\).

Now, substitute these into the formula:
\[
\text{Proy}_{\vec{b}}\vec{a} = \frac{23}{37} \vec{b} = \frac{23}{37} (6, 1) = \left(\frac{138}{37}, \frac{23}{37}\right)
\]

Simplifying the fractions:
\[
\text{Proy}_{\vec{b}}\vec{a} \approx (3.73, 0.62)
\]

### Step 2: Find the component of \(\vec{a}\) perpendicular to \(\vec{b}\)
The component of \(\vec{a}\) perpendicular to \(\vec{b}\) is given by:
\[
\text{Comp}_{\vec{b}^{\perp}}\vec{a} = \vec{a} - \text{Proy}_{\vec{b}}\vec{a}
\]

Substitute \(\vec{a} = (3, 5)\) and \(\text{Proy}_{\vec{b}}\vec{a} \approx (3.73, 0.62)\):
\[
\text{Comp}_{\vec{b}^{\perp}}\vec{a} = (3, 5) - (3.73, 0.62) = (-0.73, 4.38)
\]

### Summary
- The projection of \(\vec{a}\) onto \(\vec{b}\) is approximately \((3.73, 0.62)\).
- The component of \(\vec{a}\) perpendicular to \(\vec{b}\) is approximately \((-0.73, 4.38)\).

Would you like a more detailed explanation, or do you have any questions about these steps?

### Relative Questions
1. What is the geometric interpretation of the vector projection?
2. How do the dot product and magnitude of vectors relate to their projections?
3. Can the projection of a vector onto another be longer than the original vector?
4. What happens to the perpendicular component if the vectors are parallel?
5. How can we find the angle between two vectors using their dot product?

### Tip
When working with vector projections, remember that the projection lies in the direction of the vector onto which you are projecting, while the perpendicular component is orthogonal to this direction.

Si \( \vec{a} = (3,5) \) y \( \vec{b} = (6,1) \), representa gráficamente y determina la \text{Proy}_{\vec{b}}\vec{a} y la \text{Comp}_{\vec{b}^{\perp}}\vec{a}.

Solve vector projection and perpendicular component problems for vectors \( \vec{a} = (3,5) \) and \( \vec{b} = (6,1) \). Learn to calculate both the projection of one vector onto another and the component that is perpendicular to the second vector. This exercise covers vector mathematics and is suitable for high school students (Grades 10-12).

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