7.1 Q-9

Question Statement

Given that $O$ is the origin and $O \vec{P} = A \vec{B}$, find the point $P$ when the coordinates of points $A$ and $B$ are given as $A(-3, 7)$ and $B(1, 0)$ respectively.

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Background and Explanation

In this problem, we need to find the coordinates of point $P$ given that the vector from the origin $O$ to $P$ is equal to the vector from point $A$ to point $B$. This means that $O \vec{P} = A \vec{B}$, which gives us a relationship between the points.

To find the vector from one point to another, we use the formula for vector subtraction: $\mathbf{P Q} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j}$

Where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of points $P$ and $Q$, respectively.

Once we find the vector $A \vec{B}$, we can equate it to $O \vec{P}$ and solve for the coordinates of $P$.

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Solution

1. Set up the equation:
   The vector $O \vec{P}$ is simply the coordinates of $P$ since $O$ is the origin $(0, 0)$. The vector $A \vec{B}$ is calculated by subtracting the coordinates of $A(-3, 7)$ from $B(1, 0)$.

   Given that $O \vec{P} = A \vec{B}$, we have: $(x, y) - (0, 0) = (1, 0) - (-3, 7)$

2. Simplify the equation:
   Subtract the coordinates: $(x, y) = (1 + 3, 0 - 7)$

3. Solve for $x$ and $y$:
   Simplifying further: $(x, y) = (4, -7)$

Thus, the coordinates of point $P$ are $(4, -7)$.

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Key Formulas or Methods Used

• Vector between two points:
  The vector from point $A(x_1, y_1)$ to point $B(x_2, y_2)$ is given by: $A \vec{B} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j}$

• Vector from the origin:
  The vector from the origin $O(0, 0)$ to point $P(x, y)$ is simply: $O \vec{P} = (x, y)$

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Summary of Steps

1. Write the equation $O \vec{P} = A \vec{B}$.
2. Find $A \vec{B}$ by subtracting the coordinates of $A$ from $B$.
3. Solve for $x$ and $y$ using the resulting equation.
4. The coordinates of point $P$ are $(4, -7)$.