Precalculus Vectors in the Plane Worksheet Solutions (PDF)

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

Cover of the book Precalculus Math Worksheets with Answers (PDF)

Download as a PDFThe Precalculus worksheets book with answersPrintable PDFSee the book ›

Sommaire

This vectors practice solutions 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.

Use this key to check each vector computation. Solutions show components, magnitudes and angles step by step, with final answers in bold.

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 practice solutions: exercises 1 to 10

Exercise 1 – Component form from two points

\(\langle7-2,\ -1-3\rangle\).

\(\langle5,-4\rangle\)

Exercise 2 – Magnitude

\(\sqrt{25+144}=\sqrt{169}\).

Exercise 3 – Direction angle

\(|v|=\sqrt{4+12}=4\). The point is in quadrant II; the reference angle is \(\arctan\frac{2\sqrt3}{2}=\arctan\sqrt3=60^{\circ}\), so \(\theta=180^{\circ}-60^{\circ}\).

Magnitude \(4\); direction \(120^{\circ}\)

Exercise 4 – Components from magnitude and angle

\(\langle10\cos30^{\circ},\ 10\sin30^{\circ}\rangle=\langle5\sqrt3,\ 5\rangle\approx\langle 8.66,\ 5\rangle\).

\(\langle5\sqrt3,5\rangle\approx\langle 8.66,5\rangle\)

Exercise 5 – Add vectors

The sum goes from the first tail to the last head.

\(\langle3-1,\ -2+5\rangle\).

\(\langle2,3\rangle\)

Exercise 6 – Subtract vectors

\(\langle6-2,\ 4-7\rangle\).

\(\langle4,-3\rangle\)

Exercise 7 – Linear combination

\(\langle6,-3\rangle-\langle-2,8\rangle=\langle8,-11\rangle\).

\(\langle8,-11\rangle\)

Exercise 8 – Unit vector

\(|v|=10\). Divide each component by 10.

\(\left\langle\frac35,\frac45\right\rangle\)

Exercise 9 – i and j form

\(4\vec{i}-3\vec{j}\). \(\sqrt{16+9}=5\).

\(4\vec{i}-3\vec{j}\); \(5\)

Exercise 10 – Dot product

\(2(-3)+5(4)=-6+20\).

Exercises 11 and up

Exercise 11 – Angle between vectors

Angle of about 81.9 degrees.

\(\vec{u}\cdot\vec{v}=3-2=1\). \(|\vec{u}|=\sqrt5\), \(|\vec{v}|=\sqrt{10}\). \(\cos\theta=\frac{1}{\sqrt{50}}\approx0.1414\), so \(\theta\approx 81.9^{\circ}\).

\(81.9^{\circ}\)

Exercise 12 – Perpendicular check

\(5(2)+(-2)(5)=10-10=0\).

Yes, the dot product is 0

Exercise 13 – Find k

The dot product must be zero: \(8(3)+12k=0\), so \(k=-2\).

\(k=-2\)

Exercise 14 – Parallel vectors

\(\langle-9,6\rangle=-3\langle3,-2\rangle\). One is a scalar multiple of the other, pointing the opposite way.

Parallel (opposite directions)

Exercise 15 – Two perpendicular forces

The resultant is the diagonal.

\(\langle30,40\rangle\): magnitude \(\sqrt{900+1600}=50\). Direction \(\arctan\frac{40}{30}\approx 53.13^{\circ}\).

\(50\) N at \(53.13^{\circ}\) north of east

Exercise 16 – Airplane and wind

Ground velocity ⟨40, 250⟩.

Ground velocity \(\langle40,250\rangle\). Speed \(\sqrt{1600+62500}\approx 253.2\). Angle east of north: \(\arctan\frac{40}{250}\approx 9.1^{\circ}\).

\(253.2\) mph; \(9.1^{\circ}\) east of north

Exercise 17 – Boat in a current

Speed 6.5 mph.

Speed \(\sqrt{36+6.25}=6.5\). Drift \(\arctan\frac{2.5}{6}\approx 22.62^{\circ}\).

\(6.5\) mph; \(22.62^{\circ}\)

Exercise 18 – Work

\(30(6)+40(2)=180+80\).

\(260\) foot-pounds

Exercise 19 – Angle between two forces

\(\cos\theta=\frac{15}{5\cdot5}=0.6\), so \(\theta\approx 53.13^{\circ}\). Resultant \(\langle8,4\rangle\): \(\sqrt{64+16}=\sqrt{80}\approx8.94\).

\(53.13^{\circ}\); \(8.94\) N

Exercise 20 – Parallelogram

\(\vec{AB}=\langle3,1\rangle\) and \(\vec{DC}=\langle3,1\rangle\) are equal, so \(ABCD\) is a parallelogram. \(\vec{AC}=\langle4,4\rangle\), \(|AC|=\sqrt{32}\approx 5.66\).

\(|AC|=4\sqrt2\approx 5.66\)

Key facts behind this vectors practice solutions

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 practice solutions

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

Download the vectors practice solutions as a PDF with the button below and keep it with your notes.

All Precalculus chapters: Precalculus math.

Voter.. post

Free printable PDF