7.1 Q-11

Question Statement

Given that $A \vec{B} = C \vec{D}$, find the coordinates of point $A$ when the coordinates of points $B(1, 2)$, $C(-2, 5)$, and $D(4, 11)$ are known.

---

Background and Explanation

In this problem, we are given the relationship $A \vec{B} = C \vec{D}$, meaning the vector from point $A$ to point $B$ is equal to the vector from point $C$ to point $D$.

To solve for the coordinates of point $A$, we use the vector equation: $A \vec{B} = C \vec{D}$

The vector $A \vec{B}$ is simply the difference between the coordinates of $B$ and $A(x, y)$, and similarly, the vector $C \vec{D}$ is the difference between the coordinates of $D(4, 11)$ and $C(-2, 5)$.

We can then equate the components of these vectors and solve for $x$ and $y$.

---

Solution

Step 1: Set up the vector equation

We are given that $A \vec{B} = C \vec{D}$. This means: $(1, 2) - (x, y) = (4, 11) - (-2, 5)$

This gives us the equation to solve for $A$.

Step 2: Simplify the equation

1. Subtract the coordinates:

   • For $A \vec{B}$, we subtract $(x, y)$ from $(1, 2)$, so: $(1 - x, 2 - y)$

   • For $C \vec{D}$, we subtract $(-2, 5)$ from $(4, 11)$, so: $(4 + 2, 11 - 5) = (6, 6)$

So, we now have the equation: $(1 - x, 2 - y) = (6, 6)$

Step 3: Solve for $x$ and $y$

By comparing the components:

• $1 - x = 6$ gives $x = -5$
• $2 - y = 6$ gives $y = -4$

Thus, the coordinates of point $A$ are $(-5, -4)$.

---

Key Formulas or Methods Used

• Vector between two points:
  The vector from point $P(x_1, y_1)$ to point $Q(x_2, y_2)$ is given by:
  $P \vec{Q} = (x_2 - x_1, y_2 - y_1)$

• Equating vectors:
  If $A \vec{B} = C \vec{D}$, then the components of the vectors must be equal, which allows us to set up equations for $x$ and $y$.

---

Summary of Steps

1. Write the equation $A \vec{B} = C \vec{D}$ using the given coordinates.
2. Subtract the coordinates to express the vectors $A \vec{B}$ and $C \vec{D}$.
3. Equate the components of the vectors to solve for $x$ and $y$.
4. The coordinates of point $A$ are $(-5, -4)$.