Addition Of Vectors

Given Points:

To Find: Position vector

• Step 1: Find :
  • Expanding the determinant:

• Step 2: Compute :
  • Magnitude of :

• Direction Cosine:
• Simplify:

Step 1: Condition for collinearity.
- The points are collinear if the vectors and are parallel, i.e., .
Step 2: Find and .
The vector is calculated as: The vector is calculated as: Step 3: Compute .
- Using the determinant formula for the cross product: Expanding the determinant, we get the components of the cross product and set each component equal to zero to find and . After solving, we get the values: Step 4: Calculate .
Substitute and into the expression : Now, calculate the square: Final Answer: .

Step 1: Analyze the given condition.
.
Thus, , where is a scalar.
Step 2: Solve for .
Step 3: Use the given conditions.
1. : Simplify to solve for and .
2. Substituting into : Solving these, we find .
Step 4: Compute .
Final Answer: .

We are given that the vectors are coplanar. This means the scalar triple product .

Step 1: Using the projection formula.
The projection of vector onto vector is given by the formula: We are told that the projection of on is , so we have:

Step 2: Calculating the dot product . The dot product is calculated as follows: Expanding the dot product:

Step 3: Calculating the magnitude of . The magnitude of vector is: Step 4: Solving the equation.
Substitute the values of and into the projection formula: Simplify: Multiply both sides by :

Step 5: Coplanarity condition.
The vectors are coplanar, so we also have the condition: Now, using this condition and solving the system of equations, we find that .
Thus, the sum of all possible values of is , and the correct answer is option (1).