give answer in 2 step with explanation at the end of each step and final answer at the end:

Problem A
.

3
Using the vectors defined in problem A
.
2
,
compute the indicated vectors and

Question

give answer in 2 step with explanation at the end of each step and final answer at the end:

Problem A
.
 
3
Using the vectors defined in problem A
.
2
,
 ﻿compute the indicated vectors and also show how the results be obtained geometrically.
A
.
3
.
a
.
 ﻿vec
(
a
)
+
vec
(
b
)
,
vec
(
a
)
+
vec
(
b
)
=
[
3
0
]
+
[
2
3
]
=
[
5
3
]
A
.
3
.
b
.
 ﻿vec
(
d
)
-
vec
(
c
)
,
vec
(
d
)
-
vec
(
c
)
=
[
3
-
2
]
-
[
-
2
3
]
=
5
-
5
rail of
addroector hat
(
a
)
 ﻿to the beal of vector hat
(
b
)
0
-
7
Gind put the
tail of vector hat
(
b
)
2
 ﻿to the
head of vector 
a
-
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VIPSolutions · Accepted Answer

Let's break down **Problem A.3** step-by-step for each part, explaining both the algebraic (component-wise) calculation and the geometric interpretation.

---

### **A.3.a:  $\vec{a} + \vec{b}$**

#### **Step 1: Algebraic Calculation**

Given:
- $\vec{a} = \begin{bmatrix} 3 \ 0 \end{bmatrix}$
- $\vec{b} = \begin{bmatrix} 2 \ 3 \end{bmatrix}$

Add corresponding components:
$$
\vec{a} + \vec{b} = 
\begin{bmatrix} 3 \ 0 \end{bmatrix} +
\begin{bmatrix} 2 \ 3 \end{bmatrix} =
\begin{bmatrix} 3+2 \ 0+3 \end{bmatrix} =
\begin{bmatrix} 5 \ 3 \end{bmatrix}
$$

**Explanation:**  
Each component of the vectors is added to get the resulting vector.

---

#### **Step 2: Geometric Interpretation**

To add vectors geometrically:
- Place the **tail of $\vec{b}$** at the **head of $\vec{a}$**.
- The **resulting vector** ($\vec{a} + \vec{b}$) is drawn from the **tail of $\vec{a}$** to the **head of $\vec{b}$**.

**Explanation:**  
This is called the "tip-to-tail" method, and the resulting vector shows the total displacement if you follow $\vec{a}$ then $\vec{b}$.

---

**Final Answer for A.3.a:**  
$$
\boxed{
\vec{a} + \vec{b} = \begin{bmatrix} 5 \ 3 \end{bmatrix}
}
$$

---

### **A.3.b:  $\vec{d} - \vec{c}$**

#### **Step 1: Algebraic Calculation**

Given:
- $\vec{d} = \begin{bmatrix} 3 \ -2 \end{bmatrix}$
- $\vec{c} = \begin{bmatrix} -2 \ 3 \end{bmatrix}$

Subtract corresponding components:
$$
\vec{d} - \vec{c} = 
\begin{bmatrix} 3 \ -2 \end{bmatrix} -
\begin{bmatrix} -2 \ 3 \end{bmatrix} =
\begin{bmatrix} 3-(-2) \ -2-3 \end{bmatrix} =
\begin{bmatrix} 5 \ -5 \end{bmatrix}
$$

**Explanation:**  
Subtract each component of the second vector from the corresponding component of the first vector.

---

#### **Step 2: Geometric Interpretation**

To subtract vectors geometrically:
- Place both vectors' tails at the origin.
- The **vector from the head of $\vec{c}$ to the head of $\vec{d}$** represents $\vec{d} - \vec{c}$.

**Explanation:**  
This method shows the change needed to get from the end of $\vec{c}$ to the end of $\vec{d}$.

---

**Final Answer for A.3.b:**  
$$
\boxed{
\vec{d} - \vec{c} = \begin{bmatrix} 5 \ -5 \end{bmatrix}
}
$$

---

**Summary Table:**

| Part     | Calculation                                  | Result                   |
|----------|----------------------------------------------|--------------------------|
| A.3.a    | $\vec{a} + \vec{b} = [3,0] + [2,3]$        | $[5,3]$                |
| A.3.b    | $\vec{d} - \vec{c} = [3,-2] - [-2,3]$      | $[5,-5]$               |

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