Precalculus Vectors in the Plane Test Solutions and Scoring (PDF)

Precalculus math test solutions: Vectors in the Plane – Maths PDF

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This vectors test answers for Precalculus covers vectors in the plane (Common Core N-VM.A.1, N-VM.A.2, N-VM.A.3, N-VM.B.4, N-VM.B.5), with a free PDF to print.

Worked solutions and scoring notes for the 9-question vectors test. Use the scoring line under each solution to award partial credit.

Precalculus · Common Core State Standards: N-VM.A.1, N-VM.A.2, N-VM.A.3, N-VM.B.4, N-VM.B.5

Vectors test answers: worked solutions

Question 1 – Component form and magnitude [2 pts]

\(\langle2-(-3),\ -8-4\rangle=\langle5,-12\rangle\). Magnitude \(\sqrt{25+144}=13\).

\(\langle5,-12\rangle\); \(13\)

Scoring: 1 point each.

Question 2 – Magnitude and direction [2 pts]

\(\sqrt{25+25}=5\sqrt2\approx7.07\). Quadrant II with reference angle \(45^{\circ}\): \(135^{\circ}\).

\(5\sqrt2\); \(135^{\circ}\)

Scoring: 1 point each.

Question 3 – Vector arithmetic [2 pts]

\(\langle8,-2\rangle-\langle-6,9\rangle=\langle14,-11\rangle\).

\(\langle14,-11\rangle\)

Scoring: 1 point for each component.

Question 4 – Unit vector [2 pts]

Magnitude \(13\).

\(\left\langle-\frac{5}{13},\frac{12}{13}\right\rangle\)

Scoring: 1 point for the magnitude, 1 point for the vector.

Question 5 – Sum of two vectors [3 pts]

The sum is the diagonal.

(a) \(\vec{u}=\langle4,1\rangle\), \(\vec{v}=\langle1,3\rangle\). (b) \(\langle5,4\rangle\). (c) \(\sqrt{25+16}=\sqrt{41}\approx 6.40\).

\(\langle4,1\rangle,\langle1,3\rangle\); \(\langle5,4\rangle\); \(6.40\)

Scoring: 1 point for each part.

Question 6 – Perpendicular vectors [2 pts]

\(12-6k=0\), so \(k=2\).

\(k=2\)

Scoring: 1 point for the dot product, 1 point for k.

Question 7 – Angle between vectors [2 pts]

Dot product \(-2+12=10\). \(|\langle2,3\rangle|=\sqrt{13}\), \(|\langle-1,4\rangle|=\sqrt{17}\). \(\cos\theta=\frac{10}{\sqrt{221}}\), \(\theta\approx 47.73^{\circ}\).

\(47.73^{\circ}\)

Scoring: 1 point for the dot product and magnitudes, 1 point for the angle.

Question 8 – Airplane and wind [3 pts]

Ground velocity ⟨300, -50⟩.

(a) \(\langle300,-50\rangle\). (b) \(\sqrt{90000+2500}\approx 304.1\). (c) \(\arctan\frac{50}{300}\approx 9.5^{\circ}\) south of east.

\(\langle300,-50\rangle\); \(304.1\) mph; \(9.5^{\circ}\)

Scoring: 1 point for each part.

Question 9 – Work [2 pts]

\(20(4)+15(3)=80+45\).

\(125\) joules

Scoring: 1 point for the dot product, 1 point for the value.

Total score

Total

20 points

Key facts behind this vectors test answers

Key facts

Vector from P to Q: ⟨x2 – x1, y2 – y1⟩; magnitude sqrt(a^2 + b^2).
Components of magnitude r at angle theta: .
Unit vector in the direction of v: v divided by |v|.
Dot product: ⟨a, b⟩ . ⟨c, d⟩ = ac + bd; zero means perpendicular.
Resultant force or velocity is the vector sum of the parts.

How to use this vectors test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

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