Let `veca = 4hati + 5hatj - hatk, vecb = hati- 4 hatj + 5 hatk and vec c = 3hati + hatj- hatk.` Find a vector `vecd` which is perpendicular to both `vec a and vecb` and satisfying `vecd .vecc=21`

Answer

To find the vector \( \vec{d} \) that is perpendicular to both \( \vec{a} \) and \( \vec{b} \) and satisfies \( \vec{d} \cdot \vec{c} = 21 \), we can follow these steps:

### Step 1: Define the vectors
Given:
- \( \vec{a} = 4\hat{i} + 5\hat{j} - \hat{k} \)
- \( \vec{b} = \hat{i} - 4\hat{j} + 5\hat{k} \)
- \( \vec{c} = 3\hat{i} + \hat{j} - \hat{k} \)

Let \( \vec{d} = x\hat{i} + y\hat{j} + z\hat{k} \).

### Step 2: Set up the equations for perpendicularity
Since \( \vec{d} \) is perpendicular to \( \vec{a} \):
\[ \vec{d} \cdot \vec{a} = 0 \implies (x\hat{i} + y\hat{j} + z\hat{k}) \cdot (4\hat{i} + 5\hat{j} - \hat{k}) = 0 \]
This gives:
\[ 4x + 5y - z = 0 \quad \text{(Equation 1)} \]

Since \( \vec{d} \) is also perpendicular to \( \vec{b} \):
\[ \vec{d} \cdot \vec{b} = 0 \implies (x\hat{i} + y\hat{j} + z\hat{k}) \cdot (\hat{i} - 4\hat{j} + 5\hat{k}) = 0 \]
This gives:
\[ x - 4y + 5z = 0 \quad \text{(Equation 2)} \]

### Step 3: Set up the equation for the dot product with \( \vec{c} \)
We also have:
\[ \vec{d} \cdot \vec{c} = 21 \implies (x\hat{i} + y\hat{j} + z\hat{k}) \cdot (3\hat{i} + \hat{j} - \hat{k}) = 21 \]
This gives:
\[ 3x + y - z = 21 \quad \text{(Equation 3)} \]

### Step 4: Solve the system of equations
Now we have three equations:
1. \( 4x + 5y - z = 0 \) (Equation 1)
2. \( x - 4y + 5z = 0 \) (Equation 2)
3. \( 3x + y - z = 21 \) (Equation 3)

#### Step 4.1: Solve Equation 1 for \( z \)
From Equation 1:
\[ z = 4x + 5y \]

#### Step 4.2: Substitute \( z \) into Equation 2
Substituting \( z \) into Equation 2:
\[ x - 4y + 5(4x + 5y) = 0 \]
This simplifies to:
\[ x - 4y + 20x + 25y = 0 \implies 21x + 21y = 0 \implies x + y = 0 \implies y = -x \]

#### Step 4.3: Substitute \( y \) into Equation 1
Substituting \( y = -x \) into Equation 1:
\[ 4x + 5(-x) - z = 0 \implies 4x - 5x - z = 0 \implies -x - z = 0 \implies z = -x \]

#### Step 4.4: Substitute \( y \) and \( z \) into Equation 3
Substituting \( y = -x \) and \( z = -x \) into Equation 3:
\[ 3x + (-x) - (-x) = 21 \implies 3x - x + x = 21 \implies 3x = 21 \implies x = 7 \]

### Step 5: Find \( y \) and \( z \)
Now substituting \( x = 7 \):
\[ y = -x = -7, \quad z = -x = -7 \]

### Step 6: Write the vector \( \vec{d} \)
Thus, the vector \( \vec{d} \) is:
\[ \vec{d} = 7\hat{i} - 7\hat{j} - 7\hat{k} \]

### Final Answer
\[ \vec{d} = 7\hat{i} - 7\hat{j} - 7\hat{k} \]