Precalculus Vectors in the Plane Worksheet (PDF)

Precalculus math worksheet: Vectors in the Plane – Maths PDF

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This vectors practice worksheet 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.

These 20 problems practice vectors in the plane (Common Core N-VM.A.1 to N-VM.A.3 and N-VM.B.4, N-VM.B.5). You will find components and magnitudes, add and scale vectors, use unit vectors and the dot product and solve force and velocity problems.

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 worksheet: exercises 1 to 10

Exercise 1 – Component form from two points

Write the vector from \(A(2,3)\) to \(B(7,-1)\) in component form.

Exercise 2 – Magnitude

Find the magnitude of \(\langle-5,12\rangle\).

Exercise 3 – Direction angle

Find the magnitude and direction angle of \(\langle-2,2\sqrt3\rangle\).

Exercise 4 – Components from magnitude and angle

Write the vector with magnitude \(10\) and direction \(30^{\circ}\) in component form.

Exercise 5 – Add vectors

Head-to-tail addition.

Find \(\langle3,-2\rangle+\langle-1,5\rangle\) and explain the picture.

Exercise 6 – Subtract vectors

Find \(\langle6,4\rangle-\langle2,7\rangle\).

Exercise 7 – Linear combination

Find \(3\langle2,-1\rangle-2\langle-1,4\rangle\).

Exercise 8 – Unit vector

Find the unit vector in the direction of \(\langle6,8\rangle\).

Exercise 9 – i and j form

Write \(\langle4,-3\rangle\) in the form \(a\vec{i}+b\vec{j}\) and find its magnitude.

Exercise 10 – Dot product

Find \(\langle2,5\rangle\cdot\langle-3,4\rangle\).

Exercises 11 and up

Exercise 11 – Angle between vectors

Two vectors from the origin.

Find the angle between \(\langle1,2\rangle\) and \(\langle3,-1\rangle\) to the nearest tenth of a degree.

Exercise 12 – Perpendicular check

Are \(\langle5,-2\rangle\) and \(\langle2,5\rangle\) perpendicular?

Exercise 13 – Find k

Find \(k\) so that \(\langle8,k\rangle\) and \(\langle3,12\rangle\) are perpendicular.

Exercise 14 – Parallel vectors

Show that \(\langle-9,6\rangle\) and \(\langle3,-2\rangle\) are parallel.

Exercise 15 – Two perpendicular forces

Forces on a crate.

Two forces of \(30\) N east and \(40\) N north act on a crate. Find the magnitude of the resultant and its direction north of east, to the nearest hundredth of a degree.

Exercise 16 – Airplane and wind

Plane heading north with a crosswind.

An airplane flies due north at \(250\) mph relative to the air. A wind blows from the west (toward the east) at \(40\) mph. Find the ground speed and the angle east of north, to the nearest tenth.

Exercise 17 – Boat in a current

A river crossing.

Liam steers a boat straight across a river at \(6\) mph. The current flows downstream at \(2.5\) mph. Find the boat’s actual speed and the angle of drift from straight across, to the nearest hundredth of a degree.

Exercise 18 – Work

A force of \(\langle30,40\rangle\) pounds moves an object along \(\langle6,2\rangle\) feet. Find the work \(W=\vec{F}\cdot\vec{d}\).

Exercise 19 – Angle between two forces

Forces \(\vec{F}_{1}=\langle5,0\rangle\) N and \(\vec{F}_{2}=\langle3,4\rangle\) N act on a point. Find the angle between them and the magnitude of the resultant.

Exercise 20 – Parallelogram

Show that \(A(1,1)\), \(B(4,2)\), \(C(5,5)\) and \(D(2,4)\) form a parallelogram and find the length of diagonal \(AC\).

Key facts behind this vectors practice worksheet

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 worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

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

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