Angle Between Two Lines

1. Identify the direction vector of the line and normal vector of the plane:

Given the line equation , its direction vector is:

Given the plane equation , its normal vector is:

2. Compute the angle between the line and the plane:

The angle between the line and the plane satisfies:

where is the dot product, and , are the magnitudes of the vectors.

3. Calculate the dot product and magnitudes:

4. Compute :

Correct Answer:

1. Understand the problem:

We need to find the angle between the line and the line joining the points (1, 1) and (-3, 4).

2. Find the slope of the given line :

Rewrite the equation in slope-intercept form:

3. Find the slope of the line joining (1, 1) and (-3, 4):

The slope is:

4. Compute the angle between the two lines:

The formula for the angle between two lines with slopes and is:

Substituting and :

Thus, .

Correct Answer: (C)