Geometry And Vectors

Step 1: We are given: Where: - and are unit vectors (i.e., and ).

Step 1: Compute
Using the identity for vector projection, The projection formula is: Since is the component of perpendicular to , we can compute its magnitude: Since ,

Step 2: Compute
Recall that . By the cross product formula: Thus,

Step 3: Relating to Other Terms
Since , and the cross product is perpendicular to both vectors,

Step 4: Final Answer

Final Answer: (A)

Step 1: Recognize that the vectors are coplanar if the volume of the parallelepiped they form is zero. This volume is calculated using the scalar triple product, which is defined as: This equation ensures the three vectors lie in the same plane.
Step 2: Define the vectors based on the given position vectors: Step 3: Calculate the cross product . This is done by computing the determinant of the matrix formed by the unit vectors and the components of and : This determinant expands to: Step 4: Compute the dot product of with : Set this equal to 0 for the vectors to be coplanar:

Step 1: Expression for the function. We are given the function: Step 2: Differentiating the function. We differentiate the function with respect to using the quotient rule: Step 3: Solving for the critical points. We solve for the critical points by setting the numerator of equal to zero.
Step 4: Evaluating the maximum and minimum values. After evaluating the function at the critical points, we find the maximum and minimum values of to be .
Step 5: Sum of the maximum and minimum values. The sum is:

The length of the projection of on is .

Step 1: Find

Step 2: Recall Projection Formula
The projection of on is given by: Let , so: From this,

Step 3: Cross Product Magnitude Identity
By the cross product identity: Where . Since , we get: Now, Now, Equating and simplifying,

Final Answer: (C)

Step 1: Substituting the given integral
We need to evaluate: Using the identity: Using the identity: Thus, Since , we need to determine its sign.
Step 2: Evaluating
- If , then is positive.
- Thus, . Step 3: Evaluating the integral
Step 4: Conclusion
Thus, the correct answer is:

We are given two vectors and , and we are asked to find the vector , which lies along the bisector of the internal angle between and .
Step 1: The vector is along the angle bisector, and it can be found using the formula: where and are the magnitudes of the vectors and . First, calculate the magnitudes of and :
Step 2: Now, apply the formula for : Simplify each term: Step 3: Now, we need to adjust the magnitude of to . The magnitude of the vector is given as , and the magnitude of is found using: Simplifying this step-by-step calculation will show that the final correct vector is .

Step 1: Solving for matrix . Given that , we solve for : Step 2: Use the condition . We compute and set it equal to the identity matrix, which gives us the relationships between and .
Step 3: Solving for and . From the equations, we find that and .
Step 4: Compute .

Step 1: Recognize the vector identity used: where are the vectors .
Step 2: Apply this identity to the given vector expressions and rearrange to find relations between dot products: which implies simplifications based on the magnitudes and orthogonality of vectors.
Step 3: Calculate using properties of vector norms and the results obtained from step 2: Plug in known values and solve for the final magnitude.

We are given an equilateral triangle with side length . Points and are located such that: Also, vectors and are perpendicular.

Step 1: Position Vectors Setup
Let: Now place the points and as follows: Since ,

Step 2: Find Vectors and

Step 3: Perpendicular Condition
Vectors are perpendicular if their dot product is zero: Expanding the dot product: Equating to zero: Dividing everything by ,

Step 4: Final Answer
Final Answer: (A)

Step 1: Identify the vectors.
Let the lines be and , where:

Step 2: Calculate

Step 3: Calculate .

Step 4: Find the magnitude of .

Step 5: Calculate the shortest distance.
The shortest distance d is given by:



Therefore, the shortest distance between the skew lines is 9.