Vectors In Plane And Space

First, find the vector representing the line passing through points and . Since the line is parallel to , the direction vector is . Next, calculate the position of point as a point on the line. Using the parametric equation of the line: Now, calculate the vector , where : We are given that the distance between and is 5, so: Solve this equation for , and once you find , use the cross product of vectors and to compute the area of . The area is given by: The square of the area is . Thus, the answer is 140.

Step 1: Understanding the Concept:
We first find the squared magnitudes of the vectors and . The magnitude of a vector is .

Step 2: Key Formula or Approach:
1. .
2. .
3. We need to evaluate the ratio .

Step 3: Detailed Explanation:
1. Note that and .
2. The ratio of the sums involves terms of the form if we look at the individual terms, but we must sum them first.
3. Consider the relationship between and . However, a simpler observation in these types of symmetric trigonometric sums where is distributed across quadrants is that and often relate through the complement or specific properties of the denominator .
4. For the given , it can be shown through symmetry that the sum of the squares of the secants is equal to the sum of the squares of the cosecants over the full range of .
5. Therefore, the ratio .

Step 4: Final Answer:
The value of the ratio is 1.

Concept: The magnitude of a vector expression depends on the angle between the vectors. Maximum values generally occur when vectors are parallel or anti-parallel. Let the angle between and be .
Step 1:Find .}
Step 2:Find .}
Step 3:Consider extreme values of .} When vectors are parallel, Expression value When vectors are opposite, Expression value Thus the maximum value is