PDF] Adding and subtracting vectors: The problem with the arrow ... - Free Printable

Let's solve each part of this vector problem step by step.

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Part 1: Vector Subtraction from Grid Diagrams

We are given two vectors, $\vec{A}$ and $\vec{B}$, on a grid. We need to find $\vec{A} - \vec{B}$ in both cases and draw the result.

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#### Left Side (First Pair of Vectors)

Step 1: Determine $\vec{A}$ and $\vec{B}$ from the grid

Assume each square is 1 unit.

- For $\vec{A}$: Starts at some point and ends at a point that is 2 units right and 3 units down → So:
$$
\vec{A} = 2\hat{i} - 3\hat{j}
$$

- For $\vec{B}$: Starts at a point and ends at a point that is 3 units left and 2 units down → So:
$$
\vec{B} = -3\hat{i} - 2\hat{j}
$$

Step 2: Compute $\vec{A} - \vec{B}$
$$
\vec{A} - \vec{B} = (2\hat{i} - 3\hat{j}) - (-3\hat{i} - 2\hat{j}) = (2 + 3)\hat{i} + (-3 + 2)\hat{j} = 5\hat{i} - 1\hat{j}
$$

So,
$$
\vec{A} - \vec{B} = 5\hat{i} - \hat{j}
$$

Step 3: Draw the Resultant Vector

On the blank grid below the first pair:
- Start at any origin point.
- Move 5 units to the right and 1 unit down.
- Draw an arrow from start to end.

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#### Right Side (Second Pair of Vectors)

Again, read from the grid:

- $\vec{A}$: From its tail to head → goes 2 units right, 1 unit up →
$$
\vec{A} = 2\hat{i} + 1\hat{j}
$$

- $\vec{B}$: Goes 2 units right, 3 units up →
$$
\vec{B} = 2\hat{i} + 3\hat{j}
$$

Now compute:
$$
\vec{A} - \vec{B} = (2\hat{i} + 1\hat{j}) - (2\hat{i} + 3\hat{j}) = (2 - 2)\hat{i} + (1 - 3)\hat{j} = 0\hat{i} - 2\hat{j}
$$

So,
$$
\vec{A} - \vec{B} = -2\hat{j}
$$

Draw the Resultant Vector:
- A vertical vector pointing downward, 2 units long.
- No horizontal component.

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Part 2: Algebraic Vector Subtraction

Given:
$$
\vec{A} = 3\hat{i} + 3\hat{j}, \quad \vec{B} = -4\hat{i} + 3\hat{j}
$$

Compute:
$$
\vec{A} - \vec{B} = (3\hat{i} + 3\hat{j}) - (-4\hat{i} + 3\hat{j}) = (3 + 4)\hat{i} + (3 - 3)\hat{j} = 7\hat{i} + 0\hat{j}
$$

So,
$$
\vec{A} - \vec{B} = 7\hat{i}
$$

This is a horizontal vector pointing 7 units to the right.

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✔ Final Answers:

#### 1. First Pair (Grid):
$$
\vec{A} - \vec{B} = 5\hat{i} - \hat{j}
$$

#### 2. Second Pair (Grid):
$$
\vec{A} - \vec{B} = -2\hat{j}
$$

#### 3. Algebraic Case:
$$
\vec{A} - \vec{B} = 7\hat{i}
$$

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🔧 How to Draw the Results:

For each blank grid:

1. First Answer ($5\hat{i} - \hat{j}$):
- Start at origin (say, bottom-left corner).
- Go 5 squares right, 1 square down.
- Draw arrow from start to end.

2. Second Answer ($-2\hat{j}$):
- Start at origin.
- Go straight down 2 squares.
- Draw downward arrow.

3. Third Answer ($7\hat{i}$):
- This is not on a grid, but if you were to draw it: go 7 units right along x-axis.

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Let me know if you'd like a visual sketch description or help drawing them!