O(0,0),A(0,3),B(4,0) are the vertices of triangle OAB. A force `10hati` acts at B. What is the magnitude of moment of force about the vertex A?

Options

A. 0
B. 30 unit
C. 40 unit
D. 50 unit

Explanation

To find the magnitude of the moment of the force about vertex A, we can follow these steps:

### Step 1: Identify the Points and Force
We have the points:
- O(0, 0)
- A(0, 3)
- B(4, 0)

The force acting at point B is given as **F = 10 i** (which means the force has a magnitude of 10 units in the x-direction).

### Step 2: Determine the Position Vector from A to B
To find the moment of the force about point A, we first need to find the position vector **R** from A to B.

The position vector **R** can be calculated as:
\[ R = B - A = (4, 0) - (0, 3) = (4 - 0, 0 - 3) = (4, -3) \]
Thus, the position vector **R** is:
\[ R = 4 i - 3 j \]

### Step 3: Write the Force Vector
The force vector **F** is given as:
\[ F = 10 i \]

### Step 4: Calculate the Moment of the Force about Point A
The moment of the force about point A is given by the cross product of the position vector **R** and the force vector **F**:
\[ M_A = R \times F \]

Substituting the vectors:
\[ M_A = (4 i - 3 j) \times (10 i) \]

### Step 5: Calculate the Cross Product
To compute the cross product, we can use the determinant of a matrix:
\[ M_A = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 4 & -3 & 0 \\ 10 & 0 & 0 \end{vmatrix} \]

Calculating the determinant:
\[ M_A = \hat{i} \begin{vmatrix} -3 & 0 \\ 0 & 0 \end{vmatrix} - \hat{j} \begin{vmatrix} 4 & 0 \\ 10 & 0 \end{vmatrix} + \hat{k} \begin{vmatrix} 4 & -3 \\ 10 & 0 \end{vmatrix} \]

Calculating each of these determinants:
1. For **i** component:
\[ = -3 \cdot 0 - 0 \cdot 0 = 0 \]
2. For **j** component:
\[ = 4 \cdot 0 - 10 \cdot 0 = 0 \]
3. For **k** component:
\[ = 4 \cdot 0 - (-3) \cdot 10 = 0 + 30 = 30 \]

Thus, we have:
\[ M_A = 0 \hat{i} - 0 \hat{j} + 30 \hat{k} = 30 \hat{k} \]

### Step 6: Find the Magnitude of the Moment
The magnitude of the moment is:
\[ |M_A| = |30 \hat{k}| = 30 \]

### Final Answer
The magnitude of the moment of the force about vertex A is **30 units**.

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