7.1 Q-4

Question Statement

Find the sum of vectors $A \vec{B}$ and $\mathbf{C D}$, given the four points:

• $A(1, -1)$
• $B(2, 0)$
• $C(-1, 3)$
• $D(-2, 2)$

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Background and Explanation

To find the sum of vectors $A \vec{B}$ and $\mathbf{C D}$, we first need to calculate the individual vectors. The vector from point $A$ to point $B$ is written as $A \vec{B}$, and the vector from point $C$ to point $D$ is written as $C \vec{D}$. The vector between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is calculated using the formula: $\mathbf{P Q} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j}$

Once we have these two vectors, we add them together by combining their i and j components separately.

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Solution

Step 1: Calculate $A \vec{B}$

1. Find the components of $A \vec{B}$:
   The coordinates of $A(1, -1)$ and $B(2, 0)$ give us: $A \vec{B} = (2, 0) - (1, -1)$

2. Subtract the corresponding components:
   $A \vec{B} = (2 - 1, 0 + 1) = \hat{i} + \hat{j}$

So, the vector $A \vec{B} = \hat{i} + \hat{j}$ (Equation 1).

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Step 2: Calculate $C \vec{D}$

1. Find the components of $C \vec{D}$:
   The coordinates of $C(-1, 3)$ and $D(-2, 2)$ give us: $C \vec{D} = (-2, 2) - (-1, 3)$

2. Subtract the corresponding components:
   $C \vec{D} = (-2 + 1, 2 - 3) = -\hat{i} - \hat{j}$

So, the vector $C \vec{D} = -\hat{i} - \hat{j}$ (Equation 2).

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Step 3: Add $A \vec{B}$ and $C \vec{D}$

1. Add the two vectors:
   $A \vec{B} + C \vec{D} = (\hat{i} + \hat{j}) + (-\hat{i} - \hat{j})$

2. Simplify by combining the i and j components:
   $A \vec{B} + C \vec{D} = \hat{i} - \hat{i} + \hat{j} - \hat{j}$
   $A \vec{B} + C \vec{D} = 0$

Thus, the sum of the vectors is the null vector:
$A \vec{B} + C \vec{D} = \mathbf{0}$

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Key Formulas or Methods Used

• Vector between two points:
  $\mathbf{P Q} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j}$

• Vector addition:
  To add two vectors, combine their i and j components:
  $\mathbf{A} + \mathbf{B} = (x_1 + x_2)\hat{i} + (y_1 + y_2)\hat{j}$

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Summary of Steps

1. Find the vector $A \vec{B}$ by subtracting the coordinates of $A$ from $B$.
2. Find the vector $C \vec{D}$ by subtracting the coordinates of $C$ from $D$.
3. Add the two vectors by combining their i and j components.
4. Simplify to find the sum of the vectors, which in this case is the null vector $\mathbf{0}$.