Questions: Dot Product (Inner Product in R^n)

Question 1

Let u = (2, −1, 3) and v = (1, 4, 0). What is u · v?

• A. -2
• B. 2
• C. 7
• D. -7

Answer: (A)

u · v = (2)(1) + (−1)(4) + (3)(0) = 2 − 4 + 0 = −2. Each pair of corresponding components is multiplied and the products are summed. The zero from the third pair contributes nothing, illustrating that components beyond the shared dimension vanish.

Question 2

If u · v = 0, then at least one of u or v should be the zero vector.

• T. True
• F. False

Answer: False

Two nonzero vectors can have a zero dot product whenever they are orthogonal (perpendicular). For example, u = (1, 0) and v = (0, 1) are both nonzero but u · v = 0. Only the zero vector has a zero dot product with every other vector.

Question 3

The dot product of two nonzero vectors is negative. What does this tell you about the angle between them?

Model answer: The angle is obtuse — strictly between 90° and 180°.

From cos(θ) = (u · v) / (‖u‖ ‖v‖): since the norms ‖u‖ and ‖v‖ are positive, a negative dot product forces cos(θ) < 0, which means θ ∈ (90°, 180°). The vectors point more away from each other than toward each other.