If `bar(AB) = 2hat(i) + hat(j) - 3 hat(k)`, and A(1, 2, -1) is given point then corrdinates of B are________

Options

A. (3, 3, -4)
B. (-3, 3, -2)
C. (3, 3, 2)
D. (-3, 3, 4)

Answer

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

### Step 1: Define the Position Vectors
Let the coordinates of point B be \(B(x, y, z)\). The position vector of point A can be written as:
\[ \vec{A} = 1\hat{i} + 2\hat{j} - 1\hat{k} \]
The position vector of point B can be written as:
\[ \vec{B} = x\hat{i} + y\hat{j} + z\hat{k} \]

### Step 2: Use the Vector Equation
The vector \(\bar{AB}\) can be expressed as:
\[ \bar{AB} = \vec{B} - \vec{A} \]
Substituting the position vectors, we have:
\[ \bar{AB} = (x\hat{i} + y\hat{j} + z\hat{k}) - (1\hat{i} + 2\hat{j} - 1\hat{k}) \]
This simplifies to:
\[ \bar{AB} = (x - 1)\hat{i} + (y - 2)\hat{j} + (z + 1)\hat{k} \]

### Step 3: Set Up the Equation
According to the problem, we know that:
\[ \bar{AB} = 2\hat{i} + \hat{j} - 3\hat{k} \]
Now we can set the two expressions for \(\bar{AB}\) equal to each other:
\[ (x - 1)\hat{i} + (y - 2)\hat{j} + (z + 1)\hat{k} = 2\hat{i} + \hat{j} - 3\hat{k} \]

### Step 4: Compare Coefficients
By comparing the coefficients of \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\), we can derive the following equations:
1. \(x - 1 = 2\)
2. \(y - 2 = 1\)
3. \(z + 1 = -3\)

### Step 5: Solve for x, y, and z
Now we can solve these equations one by one:

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

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

3. From \(z + 1 = -3\):
\[ z = -3 - 1 = -4 \]

### Step 6: Write the Coordinates of B
Thus, the coordinates of point B are:
\[ B(3, 3, -4) \]

### Final Answer
The coordinates of B are \( (3, 3, -4) \).

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