Resolve a force F = 10N along x and y-axes. Where this force vector in making an angle of `60^@` from negative x-axis towards nagetive y-axis?

Explanation

To resolve the force vector \( F = 10 \, \text{N} \) along the x and y axes, we need to follow these steps:

### Step 1: Understand the angle and direction
The force vector \( F \) makes an angle of \( 60^\circ \) from the negative x-axis towards the negative y-axis. This means that the angle with the positive x-axis is \( 180^\circ - 60^\circ = 120^\circ \).

### Step 2: Identify the components of the force
To resolve the force into its x and y components, we use the following formulas:
- The x-component \( F_x = F \cos(\theta) \)
- The y-component \( F_y = F \sin(\theta) \)

Where \( \theta \) is the angle made with the positive x-axis.

### Step 3: Calculate the x-component
Using the angle \( \theta = 120^\circ \):
\[ F_x = F \cos(120^\circ) = 10 \cos(120^\circ) \]
Since \( \cos(120^\circ) = -\frac{1}{2} \):
\[ F_x = 10 \left(-\frac{1}{2}\right) = -5 \, \text{N} \]

### Step 4: Calculate the y-component
Now, calculate the y-component:
\[ F_y = F \sin(120^\circ) = 10 \sin(120^\circ) \]
Since \( \sin(120^\circ) = \frac{\sqrt{3}}{2} \):
\[ F_y = 10 \left(\frac{\sqrt{3}}{2}\right) = 5\sqrt{3} \, \text{N} \]

### Step 5: Write the components in vector form
Now, we can express the force vector \( \mathbf{F} \) in component form:
\[ \mathbf{F} = F_x \hat{i} + F_y \hat{j} = -5 \hat{i} - 5\sqrt{3} \hat{j} \]

### Final Result
Thus, the resolved components of the force \( F \) are:
\[ \mathbf{F} = -5 \hat{i} - 5\sqrt{3} \hat{j} \, \text{N} \]

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