If `vec(AB) =3 hati+ 2 hatj- hatk` and the coordinate of A are `(4,1,1),` then find the coordinates of B.

Answer

To find the coordinates of point B given the vector \(\vec{AB} = 3 \hat{i} + 2 \hat{j} - \hat{k}\) and the coordinates of point A as \((4, 1, 1)\), we can follow these steps:

### Step 1: Understand the relationship between points A and B
The vector \(\vec{AB}\) can be expressed in terms of the position vectors of points A and B. The formula is:
\[ \vec{AB} = \vec{B} - \vec{A} \]
where \(\vec{A}\) is the position vector of point A and \(\vec{B}\) is the position vector of point B.

### Step 2: Write the position vector of point A
The coordinates of point A are given as \((4, 1, 1)\). Thus, the position vector of point A is:
\[ \vec{A} = 4 \hat{i} + 1 \hat{j} + 1 \hat{k} \]

### Step 3: Write the position vector of point B
Let the coordinates of point B be \((x, y, z)\). Therefore, the position vector of point B can be expressed as:
\[ \vec{B} = x \hat{i} + y \hat{j} + z \hat{k} \]

### Step 4: Set up the equation using the vector relationship
Now, substituting the position vectors into the equation for \(\vec{AB}\):
\[ \vec{AB} = \vec{B} - \vec{A} = (x \hat{i} + y \hat{j} + z \hat{k}) - (4 \hat{i} + 1 \hat{j} + 1 \hat{k}) \]
This simplifies to:
\[ \vec{AB} = (x - 4) \hat{i} + (y - 1) \hat{j} + (z - 1) \hat{k} \]

### Step 5: Set the vectors equal to each other
Since we know that \(\vec{AB} = 3 \hat{i} + 2 \hat{j} - \hat{k}\), we can equate the components:
\[ (x - 4) \hat{i} + (y - 1) \hat{j} + (z - 1) \hat{k} = 3 \hat{i} + 2 \hat{j} - 1 \hat{k} \]

### Step 6: Create a system of equations
From the above equation, we can derive the following system of equations:
1. \(x - 4 = 3\)
2. \(y - 1 = 2\)
3. \(z - 1 = -1\)

### Step 7: Solve for x, y, and z
Now, we will solve each equation:
1. From \(x - 4 = 3\):
\[ x = 3 + 4 = 7 \]

2. From \(y - 1 = 2\):
\[ y = 2 + 1 = 3 \]

3. From \(z - 1 = -1\):
\[ z = -1 + 1 = 0 \]

### Step 8: Write the coordinates of point B
Thus, the coordinates of point B are:
\[ B(7, 3, 0) \]

### Summary
The coordinates of point B are \((7, 3, 0)\).

---