https://file.aiquickdraw.com/mathbot/file/17294676162894wj9tj4w.jpg

Let's walk through each part of the problem in the image:

---

### Part a.  
Given the coordinate vectors:  
\[
\vec{a} = \langle 3, 3 \rangle, \quad \vec{b} = \langle -2, 1 \rangle, \quad \vec{c} = \langle 2, -1 \rangle
\]

These vectors are already identified with their coordinate values.

---

### Part b.  
You are asked to **draw the vector sum** \( \vec{a} + \vec{b} + \vec{c} \) and verify your answer through vector addition.

**Adding the vectors:**
\[
\vec{a} + \vec{b} + \vec{c} = \langle 3, 3 \rangle + \langle -2, 1 \rangle + \langle 2, -1 \rangle
\]

Perform the component-wise addition:
\[
(3 - 2 + 2, 3 + 1 - 1) = \langle 3, 3 \rangle
\]

Thus, the result is:
\[
\boxed{\langle 3, 3 \rangle}
\]

---

### Part c.  
You are asked to find the sum \( 2\vec{a} + 3\vec{b} - \vec{c} \) and verify your answer.

**Calculating the scalar products:**
\[
2\vec{a} = 2 \cdot \langle 3, 3 \rangle = \langle 6, 6 \rangle
\]
\[
3\vec{b} = 3 \cdot \langle -2, 1 \rangle = \langle -6, 3 \rangle
\]
\[
-\vec{c} = -\langle 2, -1 \rangle = \langle -2, 1 \rangle
\]

**Adding these vectors:**
\[
2\vec{a} + 3\vec{b} - \vec{c} = \langle 6, 6 \rangle + \langle -6, 3 \rangle + \langle -2, 1 \rangle
\]

Component-wise:
\[
(6 - 6 - 2, 6 + 3 + 1) = \langle -2, 10 \rangle
\]

Thus, the result is:
\[
\boxed{\langle -2, 10 \rangle}
\]

---

### Part d.  
Find the sum \( x_1 \vec{a} + x_2 \vec{b} + x_3 \vec{c} \) using the given scalars:
\[
x_1 = -2, \quad x_2 = 4, \quad x_3 = -1
\]

**Calculating the scalar products:**
\[
-2\vec{a} = -2 \cdot \langle 3, 3 \rangle = \langle -6, -6 \rangle
\]
\[
4\vec{b} = 4 \cdot \langle -2, 1 \rangle = \langle -8, 4 \rangle
\]
\[
-1\vec{c} = -1 \cdot \langle 2, -1 \rangle = \langle -2, 1 \rangle
\]

**Adding these vectors:**
\[
x_1 \vec{a} + x_2 \vec{b} + x_3 \vec{c} = \langle -6, -6 \rangle + \langle -8, 4 \rangle + \langle -2, 1 \rangle
\]

Component-wise:
\[
(-6 - 8 - 2, -6 + 4 + 1) = \langle -16, -1 \rangle
\]

Thus, the result is:
\[
\boxed{\langle -16, -1 \rangle}
\]

---

### Part e.  
Find a vector \( \vec{d} \) such that:
\[
\vec{a} + \vec{c} + \vec{d} = \vec{0}
\]

**Rearrange the equation:**
\[
\vec{d} = -(\vec{a} + \vec{c})
\]

First, add \( \vec{a} + \vec{c} \):
\[
\vec{a} + \vec{c} = \langle 3, 3 \rangle + \langle 2, -1 \rangle = \langle 5, 2 \rangle
\]

Now negate the result:
\[
\vec{d} = -\langle 5, 2 \rangle = \langle -5, -2 \rangle
\]

Thus, the vector \( \vec{d} \) is:
\[
\boxed{\langle -5, -2 \rangle}
\]

---

### Summary of Answers
- **b.** \( \langle 3, 3 \rangle \)
- **c.** \( \langle -2, 10 \rangle \)
- **d.** \( \langle -16, -1 \rangle \)
- **e.** \( \langle -5, -2 \rangle \)

---

Would you like further explanations on any part of these calculations?

---

**Related Questions:**
1. How do you find the magnitude of a vector?
2. What is the geometric interpretation of adding vectors?
3. How do scalar multiples affect a vector’s direction?
4. What are some applications of vector addition in physics?
5. How does the dot product differ from vector addition?

**Tip:** When adding vectors graphically, make sure to align them tip-to-tail to visualize the resulting vector.

Find the coordinate vector for each vector in the figure and perform various vector operations including addition, scalar multiplication, and solving for an unknown vector such that the sum is zero.

Explore detailed solutions for vector operations, including adding vectors, scaling vectors, and finding unknown vectors such that the sum results in a zero vector. This guide covers the key concepts of vector addition, scalar multiplication, and solving vector equations, perfect for students in Grades 10-12.

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