7.1 Q-1

Question Statement

Write the vectors $\mathbf{P Q}$ in the form $\mathbf{x i} + \mathbf{y j}$ for the following points:

(i) $\quad \mathrm{P}(2,3), \mathrm{Q}(6,-2)$
(ii) $\quad \mathrm{P}(0,5), \mathrm{Q}(-1,-6)$

---

Background and Explanation

To solve this problem, we need to find the vector between two points. A vector is a quantity that has both magnitude and direction. The vector from point P to point Q can be calculated by subtracting the coordinates of P from the coordinates of Q.

The general form for a vector in two dimensions is:
$\mathbf{PQ} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j}$
where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of points P and Q, respectively, and $\hat{i}$, $\hat{j}$ are the unit vectors in the x and y directions.

---

Solution

(i) $\quad \mathrm{P}(2,3), \mathrm{Q}(6,-2)$

1. Find the components of the vector:

   • The vector from point P to point Q is calculated as:
     $\mathbf{PQ} = (6, -2) - (2, 3)$

2. Subtract the corresponding coordinates:

   • Subtract the x-components:
     $6 - 2 = 4$
   • Subtract the y-components:
     $-2 - 3 = -5$

3. Write the vector in unit vector form:

   • The vector becomes:
     $\mathbf{PQ} = 4\hat{i} - 5\hat{j}$

Thus, the vector from P to Q is:
$\mathbf{PQ} = 4\hat{i} - 5\hat{j}$

---

(ii) $\quad \mathrm{P}(0,5), \mathrm{Q}(-1,-6)$

1. Find the components of the vector:

   • The vector from point P to point Q is:
     $\mathbf{PQ} = (-1, -6) - (0, 5)$

2. Subtract the corresponding coordinates:

   • Subtract the x-components:
     $-1 - 0 = -1$
   • Subtract the y-components:
     $-6 - 5 = -11$

3. Write the vector in unit vector form:

   • The vector becomes:
     $\mathbf{PQ} = -1\hat{i} - 11\hat{j}$

Thus, the vector from P to Q is:
$\mathbf{PQ} = -1\hat{i} - 11\hat{j}$

---

Key Formulas or Methods Used

• Vector from P to Q:
  $\mathbf{PQ} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j}$
  Where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of points P and Q, respectively.

---

Summary of Steps

1. Identify the coordinates of points P and Q.
2. Subtract the coordinates of P from Q to find the vector components.
3. Express the vector in the form of unit vectors $\hat{i}$ and $\hat{j}$.
4. Simplify the vector expression if needed.