A vector has components (3, 4). What is its magnitude?
A7
B5
C√7
D25
Magnitude = √(3² + 4²) = √(9 + 16) = √25 = 5. This is the classic 3-4-5 right triangle. A common error is adding the components directly (3 + 4 = 7) rather than applying the Pythagorean theorem.
Question 2 True / False
A vector with components (-4, 3) and a vector with components (4, -3) are equal vectors.
TTrue
FFalse
Answer: False
Equal vectors must have both the same magnitude AND the same direction. These two vectors have the same magnitude (√(16+9) = 5) but point in opposite directions, so they are not equal. This tests the critical distinction between a scalar (magnitude only) and a vector (magnitude plus direction).
Question 3 Short Answer
A force of magnitude 10 N acts at 30° above the horizontal. What are its horizontal and vertical components?
Think about your answer, then reveal below.
Model answer: Horizontal: 10·cos(30°) = 5√3 ≈ 8.66 N. Vertical: 10·sin(30°) = 5 N.
Component form comes directly from trigonometry: the horizontal component is the adjacent side of the right triangle formed by the vector, so it uses cosine. The vertical component is the opposite side, so it uses sine. This conversion between magnitude-direction form and component form is the core skill of the topic.