Vectors

Step 1: Square both sides. | A+ B|²=| A- B|²
Step 2: Simplify. A· B=0

Using vector identity C² = A² + B² + 2ABcosθ the angle depends on the sign of cosθ, determining whether it is acute, obtuse, or right.

Step 1: Using vector formula R² = A² + B² + 2ABcosθ
Step 2: Given A=B=R R² = R² + R² + 2R²cosθ
Step 3: 1 = 2(1+cosθ) ⟹ cosθ=\frac12 ⟹ θ=(π)/(3)

Step 1: Resultant vector: vecR = vecP + vecQ = 2hati -2hatj +2hatk
Step 2: Magnitude: |vecR| = √(2² + (-2)² + 2²) = √(12) = 3√(3)

Step 1: Understanding the Question:
A vector is rotated in a 2D plane. Rotation changes the direction but leaves the magnitude of the vector unchanged.

Step 2: Key Formula or Approach:
1. Magnitude .
2. Initial angle .

Step 3: Detailed Explanation:
Let the initial vector be .
Magnitude: .
Initial angle with positive x-axis: .
The vector is rotated 30 clockwise.
New angle with x-axis = .
The new vector has magnitude 2 and is directed along the positive x-axis.
New Vector = .

Step 4: Final Answer:
The new vector is .