The force respresented by `3hat(i)+2hat(k)` is acting through the point `5hat(i)+4hat(j)-3hat(k)`. Find the moment about the point `hat(i)+3hat(j)+hat(k)`.

Answer

To solve the problem step by step, we need to find the moment of the force about a given point. Here’s the detailed solution:

### Step 1: Identify the Given Information
- Force vector \( \mathbf{F} = 3\hat{i} + 2\hat{k} \)
- Point where the force is acting \( \mathbf{A} = 5\hat{i} + 4\hat{j} - 3\hat{k} \)
- Point about which we need to find the moment \( \mathbf{B} = \hat{i} + 3\hat{j} + \hat{k} \)

### Step 2: Calculate the Position Vector \( \mathbf{R} \)
The position vector \( \mathbf{R} \) from point \( \mathbf{B} \) to point \( \mathbf{A} \) is given by:
\[ \mathbf{R} = \mathbf{A} - \mathbf{B} \]
Substituting the values:
\[ \mathbf{R} = (5\hat{i} + 4\hat{j} - 3\hat{k}) - (\hat{i} + 3\hat{j} + \hat{k}) \]
Calculating each component:
\[ \mathbf{R} = (5 - 1)\hat{i} + (4 - 3)\hat{j} + (-3 - 1)\hat{k} = 4\hat{i} + 1\hat{j} - 4\hat{k} \]

### Step 3: Calculate the Moment \( \mathbf{M} \)
The moment \( \mathbf{M} \) about point \( \mathbf{B} \) is given by the cross product:
\[ \mathbf{M} = \mathbf{R} \times \mathbf{F} \]
Substituting the vectors:
\[ \mathbf{M} = (4\hat{i} + 1\hat{j} - 4\hat{k}) \times (3\hat{i} + 2\hat{k}) \]

### Step 4: Set Up the Determinant for the Cross Product
Using the determinant to calculate the cross product:
\[ \mathbf{M} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 4 & 1 & -4 \\ 3 & 0 & 2 \end{vmatrix} \]

### Step 5: Calculate the Determinant
Calculating the determinant:
1. For \( \hat{i} \):
\[ \hat{i} \cdot (1 \cdot 2 - 0 \cdot -4) = 2\hat{i} \]
2. For \( \hat{j} \):
\[ -\hat{j} \cdot (4 \cdot 2 - 3 \cdot -4) = - (8 + 12) = -20\hat{j} \]
3. For \( \hat{k} \):
\[ \hat{k} \cdot (4 \cdot 0 - 3 \cdot 1) = -3\hat{k} \]

Combining these results:
\[ \mathbf{M} = 2\hat{i} - 20\hat{j} - 3\hat{k} \]

### Step 6: Calculate the Magnitude of the Moment
The magnitude of the moment vector is given by:
\[ |\mathbf{M}| = \sqrt{(2)^2 + (-20)^2 + (-3)^2} \]
Calculating:
\[ |\mathbf{M}| = \sqrt{4 + 400 + 9} = \sqrt{413} \]

### Final Result
The moment about the point \( \mathbf{B} \) is:
\[ \mathbf{M} = 2\hat{i} - 20\hat{j} - 3\hat{k} \]
And the magnitude of the moment is:
\[ |\mathbf{M}| = \sqrt{413} \]

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