https://r2.erweima.ai/mathbot/image/1726590969463g4a60kis.jpg

The problem asks to graphically represent and determine the projection of vector **\(\vec{a}\)** onto vector **\(\vec{b}\)**, as well as the component of **\(\vec{a}\)** perpendicular to **\(\vec{b}\)**.

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

We are tasked with finding:
1. **Projection of \(\vec{a}\) onto \(\vec{b}\)**: \(\text{Proj}_{\vec{b}}\vec{a}\)
2. **Perpendicular component of \(\vec{a}\) relative to \(\vec{b}\)**: \(\text{Comp}_{\vec{b}^\perp}\vec{a}\)

### Step 1: Projection Formula
The projection of \(\vec{a}\) onto \(\vec{b}\) is given by:

\[
\text{Proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{\vec{b} \cdot \vec{b}} \vec{b}
\]

Let's first calculate the dot products:
- \(\vec{a} \cdot \vec{b} = 3(6) + 5(1) = 18 + 5 = 23\)
- \(\vec{b} \cdot \vec{b} = 6(6) + 1(1) = 36 + 1 = 37\)

Now, we can calculate the projection:

\[
\text{Proj}_{\vec{b}}\vec{a} = \frac{23}{37} \cdot \vec{b} = \frac{23}{37} \cdot (6, 1)
\]

This results in:

\[
\text{Proj}_{\vec{b}}\vec{a} = \left( \frac{138}{37}, \frac{23}{37} \right) = (3.73, 0.62)
\]

### Step 2: Perpendicular Component
The perpendicular component is the vector difference:

\[
\text{Comp}_{\vec{b}^\perp}\vec{a} = \vec{a} - \text{Proj}_{\vec{b}}\vec{a}
\]

Now subtract the projection from \(\vec{a}\):

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

### Final Answers:
- \(\text{Proj}_{\vec{b}}\vec{a} = (3.73, 0.62)\)
- \(\text{Comp}_{\vec{b}^\perp}\vec{a} = (-0.73, 4.38)\)

Would you like further details, or a graphical representation?

#### Related Questions:
1. What is the geometrical meaning of the projection of a vector?
2. How is the dot product used in vector projections?
3. What is the significance of the perpendicular component of a vector?
4. How does changing the direction of vector \(\vec{b}\) affect the projection?
5. Can projections be used in applications like physics or engineering?

**Tip:** When calculating projections, always remember to normalize the direction vector if it isn’t already normalized.

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

This problem explores the projection of the vector (3,5) onto (6,1), and the perpendicular component calculation. Using vector projection formulas and dot products, the solution is calculated step-by-step. Suitable for students studying Linear Algebra.

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