Given `p=3hat(i)+2hat(j)+4hat(k), a=hat(i)+hat(j), b=hat(j)+hat(k), c=hat(i)+hat(k)` and `p=x a +y b +z c`, then x, y and z are respectively

Options

A. `3/2, 1/2, 5/2`
B. `1/2, 3/2, 5/2`
C. `5/2, 3/2, 1/2`
D. `1/2, 5/2, 3/2`

Explanation

To solve the problem, we need to express the vector \( p \) in terms of the vectors \( a \), \( b \), and \( c \). We are given:

\[ p = 3\hat{i} + 2\hat{j} + 4\hat{k} \]
\[ a = \hat{i} + \hat{j} \]
\[ b = \hat{j} + \hat{k} \]
\[ c = \hat{i} + \hat{k} \]

We need to find \( x \), \( y \), and \( z \) such that:

\[ p = x a + y b + z c \]

### Step 1: Substitute the vectors into the equation

Substituting the expressions for \( a \), \( b \), and \( c \) into the equation gives:

\[ p = x(\hat{i} + \hat{j}) + y(\hat{j} + \hat{k}) + z(\hat{i} + \hat{k}) \]

Expanding this, we have:

\[ p = x\hat{i} + x\hat{j} + y\hat{j} + y\hat{k} + z\hat{i} + z\hat{k} \]

### Step 2: Combine like terms

Now, combine the like terms:

\[ p = (x + z)\hat{i} + (x + y)\hat{j} + (y + z)\hat{k} \]

### Step 3: Set the coefficients equal

Since \( p = 3\hat{i} + 2\hat{j} + 4\hat{k} \), we can equate the coefficients of \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \):

1. \( x + z = 3 \) (1)
2. \( x + y = 2 \) (2)
3. \( y + z = 4 \) (3)

### Step 4: Solve the system of equations

Now we have a system of three equations. We can solve these equations step by step.

**Adding equations (1), (2), and (3):**

\[ (x + z) + (x + y) + (y + z) = 3 + 2 + 4 \]

This simplifies to:

\[ 2x + 2y + 2z = 9 \]

Dividing through by 2 gives:

\[ x + y + z = \frac{9}{2} \quad (4) \]

### Step 5: Substitute and solve for \( y \)

Now, we can use equation (1) to express \( z \) in terms of \( x \):

From (1):

\[ z = 3 - x \quad (5) \]

Substituting (5) into (4):

\[ x + y + (3 - x) = \frac{9}{2} \]

This simplifies to:

\[ y + 3 = \frac{9}{2} \]

Thus,

\[ y = \frac{9}{2} - 3 = \frac{9}{2} - \frac{6}{2} = \frac{3}{2} \]

### Step 6: Substitute \( y \) back to find \( x \) and \( z \)

Now substitute \( y = \frac{3}{2} \) into equation (2):

\[ x + \frac{3}{2} = 2 \]

This gives:

\[ x = 2 - \frac{3}{2} = \frac{4}{2} - \frac{3}{2} = \frac{1}{2} \]

Now substitute \( x = \frac{1}{2} \) into equation (5) to find \( z \):

\[ z = 3 - \frac{1}{2} = \frac{6}{2} - \frac{1}{2} = \frac{5}{2} \]

### Final Values

Thus, we have:

\[ x = \frac{1}{2}, \quad y = \frac{3}{2}, \quad z = \frac{5}{2} \]

### Conclusion

The values of \( x \), \( y \), and \( z \) are:

\[ \boxed{\left( \frac{1}{2}, \frac{3}{2}, \frac{5}{2} \right)} \]