Find the slope and inclination of the line through each pair of the following points:
(0, 0) and `(-sqrt(3), 3)`

Answer

To find the slope and inclination of the line through the points (0, 0) and (-√3, 3), we will follow these steps:

### Step 1: Identify the points
Let the points be:
- Point A (x1, y1) = (0, 0)
- Point B (x2, y2) = (-√3, 3)

### Step 2: Use the slope formula
The formula for the slope (m) of a line through two points is given by:
\[ m = \frac{y2 - y1}{x2 - x1} \]

### Step 3: Substitute the coordinates into the slope formula
Substituting the coordinates of points A and B into the formula:
\[ m = \frac{3 - 0}{-√3 - 0} = \frac{3}{-√3} \]

### Step 4: Simplify the slope
Now, simplify the expression:
\[ m = -\frac{3}{√3} = -\sqrt{3} \]

### Step 5: Find the inclination angle
The inclination (θ) of the line can be found using the relationship between the slope and the tangent of the angle:
\[ \tan(θ) = m \]
So,
\[ \tan(θ) = -\sqrt{3} \]

### Step 6: Determine the angle
The angle whose tangent is -√3 is:
\[ θ = 120^\circ \quad \text{(since the tangent is negative in the second quadrant)} \]

### Final Answer
Thus, the slope of the line is:
\[ \text{slope} = -\sqrt{3} \]
And the inclination of the line is:
\[ \text{inclination} = 120^\circ \]

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