If `vec(a)` position vector of a point `(1, -3)` and A is another point `(-1,5)` then what are the coordinates of the point B such that `vec(AB)= vec(a)`?

Options

A. A) `(2,0)`
B. B) `(0,2)`
C. C) `(-2,0)`
D. D) `(0,-2)`

Answer

To solve the problem step by step, we need to find the coordinates of point B such that the vector AB equals the position vector of point A.

### Step 1: Define the position vectors
Let’s denote the position vector of point A as \( \vec{A} \) and the position vector of point B as \( \vec{B} \).

Given:
- Point A has coordinates (-1, 5), so:
\[ \vec{A} = -1 \hat{i} + 5 \hat{j} \]

- Point a has coordinates (1, -3), so:
\[ \vec{a} = 1 \hat{i} - 3 \hat{j} \]

### Step 2: Express vector AB
The vector \( \vec{AB} \) can be expressed in terms of the position vectors of points A and B:
\[ \vec{AB} = \vec{B} - \vec{A} \]

Let the coordinates of point B be \( (x, y) \), so:
\[ \vec{B} = x \hat{i} + y \hat{j} \]

Now substituting \( \vec{A} \) into the equation:
\[ \vec{AB} = (x \hat{i} + y \hat{j}) - (-1 \hat{i} + 5 \hat{j}) = (x + 1) \hat{i} + (y - 5) \hat{j} \]

### Step 3: Set up the equation
According to the problem, we have:
\[ \vec{AB} = \vec{a} \]
This gives us:
\[ (x + 1) \hat{i} + (y - 5) \hat{j} = 1 \hat{i} - 3 \hat{j} \]

### Step 4: Equate coefficients
Now we can equate the coefficients of \( \hat{i} \) and \( \hat{j} \):
1. For \( \hat{i} \):
\[ x + 1 = 1 \]
Solving this gives:
\[ x = 1 - 1 = 0 \]

2. For \( \hat{j} \):
\[ y - 5 = -3 \]
Solving this gives:
\[ y = -3 + 5 = 2 \]

### Step 5: Conclusion
Thus, the coordinates of point B are:
\[ B(0, 2) \]