Questions: Dot Product

Question 1

Vectors u = (3, 0) and v = (0, 5). What is u · v, and what does it reveal about these vectors?

• A. 15 — multiply the nonzero components
• B. 8 — add all the components together
• C. 0 — the vectors are perpendicular to each other
• D. (0, 0) — the dot product cancels the vectors

Answer: (C)

u · v = 3·0 + 0·5 = 0. The dot product is always a scalar (never a vector), and a value of zero means the angle between the vectors is 90° — they are orthogonal. This is the key test for perpendicularity: you don't need to find the angle explicitly, just check whether the dot product is zero. Options A and B reflect common arithmetic errors; option D reflects the misconception that the dot product produces a vector.

Question 2

If u · v = −20 and both u and v are nonzero vectors, what can you conclude about the angle θ between them?

• A. θ = 90°, because the product is not zero
• B. θ is obtuse (greater than 90°), because cos θ must be negative
• C. θ = 180°, because the dot product is negative
• D. The vectors must have opposite signs in at least one component, so no conclusion about θ is possible

Answer: (B)

From the geometric formula u · v = |u||v|cos θ, a negative dot product means cos θ < 0, which means θ is between 90° and 180° — obtuse. The dot product is exactly −|u||v| only when θ = 180° (exactly opposite). A negative dot product rules out θ ≤ 90° but doesn't pin down the angle precisely. The sign of the dot product is a reliable indicator of which 'half' the angle falls in: positive (same general direction), zero (perpendicular), negative (opposing direction).

Question 3

The dot product of two nonzero vectors can equal zero.

• T. True
• F. False

Answer: True

Yes — when two nonzero vectors are perpendicular (θ = 90°), cos 90° = 0, so u · v = |u||v|·0 = 0. This is the orthogonality test. For example, (1, 0) · (0, 1) = 1·0 + 0·1 = 0, even though neither vector is the zero vector. This is one of the most important uses of the dot product: efficiently checking whether two vectors are perpendicular without computing any angles.

Question 4

The dot product u · v measures the length of the vector formed by adding u and v.

• T. True
• F. False

Answer: False

This confuses two entirely different operations. The dot product produces a scalar measuring directional agreement between the vectors. Adding u and v produces a new vector (by the parallelogram rule), and its length is |u + v| — a completely separate computation. The dot product encodes the angle between vectors via u · v = |u||v|cos θ, not anything about their vector sum.

Question 5

Why does the dot product equal zero when two vectors are perpendicular? Use the geometric formula to explain the connection between angle and scalar output.

Model answer: The geometric formula is u · v = |u||v|cos θ. When θ = 90°, cos 90° = 0, so the entire product collapses to zero regardless of the magnitudes. Geometrically, the dot product measures 'how much' the vectors align — how much one projects onto the other. At 90°, neither vector has any component in the direction of the other, so there is zero directional overlap, and the dot product reflects this with a value of zero.

This is why the dot product is the standard orthogonality test. The algebraic formula (sum of component products) and the geometric formula (|u||v|cos θ) are two ways of computing the same quantity, so checking whether the component sum is zero is equivalent to checking whether the angle is 90°.