Precalculus Vectors in the Plane Test Solutions and Scoring (PDF)
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.
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]
(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]
(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
20 points
Key facts behind this vectors test answers
- 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.
All Precalculus chapters: Precalculus math.
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