5 examples ADDING VECTORS BY THE GRAPHIC AND ANALYTICAL METHOD

Sure, let's work on adding vectors using both the graphical and analytical methods.

### Example 1:
Given two vectors \vec{A} = 2\hat{i} + 3\hat{j} and \vec{B} = -3\hat{i} + 4\hat{j} .

**Graphical Method:**
Draw the vectors on a coordinate system and add them head-to-tail to find the resultant vector.

**Analytical Method:**
\vec{A} + \vec{B} = (2\hat{i} + 3\hat{j}) + (-3\hat{i} + 4\hat{j}) = (2 - 3)\hat{i} + (3 + 4)\hat{j} = -\hat{i} + 7\hat{j} .

### Example 2:
Given two vectors \vec{P} = 4\hat{i} - \hat{j} and \vec{Q} = 2\hat{i} + 3\hat{j} .

**Graphical Method:**
Draw the vectors on a coordinate system and add them head-to-tail to find the resultant vector.

**Analytical Method:**
\vec{P} + \vec{Q} = (4\hat{i} - \hat{j}) + (2\hat{i} + 3\hat{j}) = (4 + 2)\hat{i} + (-1 + 3)\hat{j} = 6\hat{i} + 2\hat{j} .

### Example 3:
Given two vectors \vec{X} = -\hat{i} + 2\hat{j} and \vec{Y} = 3\hat{i} + \hat{j} .

**Graphical Method:**
Draw the vectors on a coordinate system and add them head-to-tail to find the resultant vector.

**Analytical Method:**
\vec{X} + \vec{Y} = (-\hat{i} + 2\hat{j}) + (3\hat{i} + \hat{j}) = (-1 + 3)\hat{i} + (2 + 1)\hat{j} = 2\hat{i} + 3\hat{j} .

### Example 4:
Given two vectors \vec{M} = 5\hat{i} - 3\hat{j} and \vec{N} = -2\hat{i} + \hat{j} .

**Graphical Method:**
Draw the vectors on a coordinate system and add them head-to-tail to find the resultant vector.

**Analytical Method:**
\vec{M} + \vec{N} = (5\hat{i} - 3\hat{j}) + (-2\hat{i} + \hat{j}) = (5 - 2)\hat{i} + (-3 + 1)\hat{j} = 3\hat{i} - 2\hat{j} .

### Example 5:
Given two vectors \vec{U} = 3\hat{i} + 2\hat{j} and \vec{V} = -\hat{i} + 5\hat{j} .

**Graphical Method:**
Draw the vectors on a coordinate system and add them head-to-tail to find the resultant vector.

**Analytical Method:**
\vec{U} + \vec{V} = (3\hat{i} + 2\hat{j}) + (-\hat{i} + 5\hat{j}) = (3 - 1)\hat{i} + (2 + 5)\hat{j} = 2\hat{i} + 7\hat{j} .

### Answer:
The resulting vector for each of the examples is:
1. -\hat{i} + 7\hat{j}
2. 6\hat{i} + 2\hat{j}
3. 2\hat{i} + 3\hat{j}
4. 3\hat{i} - 2\hat{j}
5. 2\hat{i} + 7\hat{j}