General Equation Of A Line

To determine the distance from point to the plane , measured parallel to a line with direction ratios , let's solve the problem step-by-step.

Step 1: Identify the given points and plane equation.

The foot of the perpendicular from point to the plane is given as .

Step 2: Understand the concept of direction ratios and vector projection.

The distance parallel to a line is concerned with projection along its direction. The direction ratios of the line are given as .

Step 3: Calculate the vector from A to its foot on the plane.

From given points and the foot of the perpendicular, , the vector from A to the foot on the plane is:

Step 4: Compute the projection of this vector along the line direction ratios.

The direction vector of the given line is .

The formula for the projection of a vector on a vector is:

Hence, the unit vector along the given line is , where:

Now, calculate the projection magnitude:

Evaluate :

Therefore,

Conclusion:

The distance of point from the plane along the specified direction is:

This corresponds to option .

To solve this problem, let's assume the following:

• The tower's height is .
• The point is directly north of the base of the tower.
• The point is 9 units west from .

Since the point is 15 units away from the tower, we can apply the Pythagorean theorem to the right triangle formed by the tower, point , and point . This is because is due north, and is due west of .

In triangle :

• units (distance between and )
• units (distance from the tower to point )
• (distance from the tower to point )

Using the Pythagorean theorem, we have:

So, the distance is 12 units. Now, using the definition of cotangent, which is the adjacent over the opposite side in the right triangle:

The angle of elevation from can be written in terms of its tangent:

The given angle at is:

Since the secant of this angle gives the distance from to :

Using trigonometric identities, the sine of this angle gives the height :

Calculate :

The cotangent of angle is:

Since , we have:

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Finally, plug back in to find :

Therefore, the cotangent of angle is , making the correct answer:

The given line passes through the point and is perpendicular to the plane . The equations of the two lines are:

• ,
• .

The shortest distance between the two skew lines is given by:

• is the vector from point on to point on ,
• is the cross product of the direction ratios of the two lines.

The vector is:

The direction ratios of are , and the direction ratios of are . The cross product is:

Expanding the determinant gives:

The magnitude of is:

Now, compute the dot product :

The shortest distance is:

Conclusion

The shortest distance between the two lines is: