Vector Basics

Since and
and

Step 1: Understanding the Concept:
Two non-zero vectors are defined as orthogonal (perpendicular to each other) if and only if their scalar (dot) product is exactly zero.

Step 2: Key Formula or Approach:
For vectors and , the orthogonality condition is: . Calculate the dot product of the given vectors and solve for the unknown parameter .

Step 3: Detailed Explanation:
Given vectors: Set their dot product to zero for orthogonality:

Step 4: Final Answer:
The value of is .

Step 1: Understanding the Concept:
The given condition signifies that the diagonals of a parallelogram formed by vectors and are equal in length. Geometrically, this means the parallelogram is a rectangle, which implies that vectors and must be perpendicular.

Step 2: Key Formula or Approach:
1. Recognize that . Alternatively, square both sides: . 2. Calculate the dot product of the given vectors and and set it to zero. 3. Solve the resulting equation to find the value of .

Step 3: Detailed Explanation:
Given vectors: As established, the condition implies that the vectors are orthogonal: Now, compute the dot product: Divide the entire equation by 2: The question asks for the value of , which we have directly found to be 1.

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