Find the magnitude of two vectors a⃗ and b⃗, having the same magnitude and such that the angle between them is 60° and their scalar product is 1/2.
Given: |a⃗| = |b⃗|, angle between them θ = 60°, and scalar product a⃗ · b⃗ = 1/2
Formula: The scalar product is defined as a⃗ · b⃗ = |a⃗| |b⃗| cos θ
Solution: Let |a⃗| = |b⃗| = x (since both vectors have same magnitude)
Substituting in the formula: x × x × cos 60° = 1/2 x² × 1/2 = 1/2 x² = 1 x = 1
Answer: |a⃗| = |b⃗| = 1
Explanation
This question tests the application of the scalar product formula from the textbook. Since both vectors have the same magnitude, we denote it by a single variable x. The formula a⃗ · b⃗ = |a⃗| |b⃗| cos θ is applied with cos 60° = 1/2 to solve for x.
Solution Steps
Step 1: Let |a⃗| = |b⃗| = x (since both vectors have same magnitude)
Step 2: Apply scalar product formula: a⃗ · b⃗ = |a⃗| |b⃗| cos θ
Step 3: Substitute: x² × cos 60° = 1/2
Step 4: Since cos 60° = 1/2: x² × 1/2 = 1/2
Step 5: Simplify: x² = 1, so x = 1
Step 6: Therefore, |a⃗| = |b⃗| = 1