Vector Algebra

k̂

By the right-hand rule for unit vectors: î × ĵ = k̂. This is a fundamental property of orthogonal unit vectors. The sequence is î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ. Common mistake: Students might confuse the direction or forget the right-hand rule.

5

The distance from origin is the magnitude of position vector: |r| = √(3² + 4²) = √(9 + 16) = √25 = 5. This forms a 3-4-5 right triangle. Common mistake: Students might add components instead of using Pythagorean theorem.

-4î + 5ĵ

First calculate: 2a = 2(î + ĵ) = 2î + 2ĵ and 3b = 3(2î - ĵ) = 6î - 3ĵ. Then 2a - 3b = (2î + 2ĵ) - (6î - 3ĵ) = 2î + 2ĵ - 6î + 3ĵ = -4î + 5ĵ. Common mistake: Sign errors when subtracting or distributing scalar multiplication.

6

Using the formula a·b = |a||b|cos θ = 3 × 4 × cos(60°) = 12 × (1/2) = 6. Remember that cos(60°) = 1/2. Common mistake: Students might use sin instead of cos or forget the cosine value.