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Vectors in the Plane Test Answer Key with Scoring – Precalculus

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Last updated: October 22, 2026 · Written by the Math Made Easy math teachers

Precalculus Math Test Answer Keys – Mathbot, the Math Made Easy robot coach, with free printable test answer keys for Precalculus
Each solution shows the work with a scoring note. Award points for correct components, for the correct operation and for the final value separately, so a sign slip costs only one point.

Standards: N-VM.A.1, N-VM.A.2, N-VM.A.3, N-VM.B.4, N-VM.B.5 · Level: Grade 12 Precalculus

Question 1 Vector Between Points (2 points)

⟨5 − (−2), −1 − 6⟩ = ⟨7, −7⟩.

Magnitude = √(49 + 49) = 7√2 ≈ 9.90.

Scoring: 1 point per part.

Question 2 Components from a Direction (2 points)

Horizontal: 40 cos 125° ≈ −22.94. Vertical: 40 sin 125° ≈ 32.77. The horizontal component is negative because 125° is in Quadrant II.

Scoring: 1 point per component.

Question 3 Vector Sum from a Figure (2 points)

1234567−3−2−112345678uvu + v
The sum goes from the first tail to the second tip.
  1. u = ⟨2, 5⟩ and v = ⟨6 − 2, 2 − 5⟩ = ⟨4, −3⟩, so u + v = ⟨6, 2⟩.
  2. 2u − v = ⟨4, 10⟩ − ⟨4, −3⟩ = ⟨0, 13⟩.

Scoring: 1 point per part.

Question 4 Unit Vector (3 points)

|v| = √58 ≈ 7.616.

(a) Unit vector = ⟨−3/√58, 7/√58⟩ ≈ ⟨−0.394, 0.919⟩.

(b) 9/58 + 49/58 = 58/58 = 1, so the magnitude is 1.

(c) 20 times the unit vector ≈ ⟨−7.88, 18.38⟩.

Scoring: 1 point per part.

Question 5 Dot Product and Angle (3 points)

(a) u · v = 5(3) + (−2)(5) = 15 − 10 = 5.

(b) |u| = √29 ≈ 5.385, |v| = √34 ≈ 5.831.

(c) cos θ = 5 ÷ (5.385 × 5.831) ≈ 0.1592, so θ ≈ 80.8°.

Scoring: 1 point per part.

Question 6 Kayak in a Current (3 points)

  1. With east as +x and north as +y: ⟨1.8, 5.5⟩.
  2. √(1.82 + 5.52) = √33.49 ≈ 5.79 mph.
  3. arctan(1.8/5.5) ≈ 18.1°.

Scoring: 1 point per part.

Question 7 Forces in Equilibrium (3 points)

−40−2020−40−202040F₁F₂F₃
The third force points opposite to the sum of the other two.
  1. F1 + F2 = ⟨−15, 38⟩. Equilibrium requires a total of ⟨0, 0⟩, so F3 = ⟨15, −38⟩.
  2. |F3| = √(225 + 1444) = √1669 ≈ 40.9 lb.

Scoring: 2 points for F3 (1 for the sum, 1 for the opposite); 1 point for the magnitude.

Question 8 Perpendicular Vectors (2 points)

Perpendicular vectors have dot product 0: 6k + (−5)(4) = 0, so 6k − 20 = 0.

k = 20/6 = 10/3.

Scoring: 1 point for the dot-product equation; 1 point for k.

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