Precalculus Vectors Worksheet with Word Problems – 22 Problems PDF

Precalculus Vectors Worksheet: problems 1–5

1. Component form from two points

Write each vector in component form and find its magnitude.

1. The vector from \(A(2, 1)\) to \(B(7, 13)\), shown below.
2. The vector from \(C(-3, 4)\) to \(D(1, -2)\).
3. The vector from \(E(5, 5)\) to the origin.

2. Magnitude of a vector

Find the magnitude of each vector.

1. \(\langle -6, 8\rangle\)
2. \(\langle 5, -12\rangle\)
3. \(\langle 9, -40\rangle\)
4. \(\langle 2, -3\rangle\)

3. Direction angles

Find the direction angle, from 0\(^{\circ}\) up to but not including 360\(^{\circ}\), of each vector in the figure.

1. \(\vec{a}=\langle 3, 3\rangle\)
2. \(\vec{b}=\langle -4, 4\rangle\)
3. \(\vec{c}=\langle -5, -5\rangle\)
4. \(\vec{d}=\langle 1, -\sqrt{3}\rangle\)

4. Components from magnitude and angle

Write the vector with the given magnitude and direction angle in component form. Give exact values.

1. Magnitude 10, angle \(30^{\circ}\)
2. Magnitude 8, angle \(120^{\circ}\)
3. Magnitude 12, angle \(270^{\circ}\)
4. Magnitude \(6\sqrt{2}\), angle \(45^{\circ}\)

5. Add and subtract vectors

Let \(\vec{u}=\langle 3, -2\rangle\) and \(\vec{v}=\langle -1, 5\rangle\). Find each result.

1. \(\vec{u}+\vec{v}\)
2. \(\vec{u}-\vec{v}\)
3. \(\vec{v}-\vec{u}\)

Precalculus Vectors Worksheet: problems 6–10

6. Scalar multiplication

Let \(\vec{u}=\langle 3, -2\rangle\), \(\vec{v}=\langle -1, 5\rangle\), \(\vec{w}=\langle 6, -10\rangle\) and \(\vec{z}=\langle 9, -6\rangle\). Find each vector.

1. \(4\vec{u}\)
2. \(-3\vec{v}\)
3. \(\frac{1}{2}\vec{w}\)
4. \(-\frac{2}{3}\vec{z}\)

7. Linear combinations

Use \(\vec{u}=\langle 3, -2\rangle\) and \(\vec{v}=\langle -1, 5\rangle\).

1. Find \(2\vec{u}-3\vec{v}\).
2. Find \(3\vec{u}+2\vec{v}\).
3. Find the magnitude of \(2\vec{u}-3\vec{v}\).

8. Unit vectors

Find the unit vector in the direction of each vector.

1. \(\langle 8, -6\rangle\)
2. \(\langle 5, 12\rangle\)
3. \(\langle -7, 24\rangle\)
4. \(\langle 0, -9\rangle\)

9. Unit vector notation i and j

Complete each part.

1. Write \(\langle 4, -3\rangle\) in terms of \(\vec{i}\) and \(\vec{j}\).
2. Write \(7\vec{i}+2\vec{j}\) in component form and find its magnitude.
3. Write \(-\vec{i}+6\vec{j}\) in component form.
4. Simplify \(3(2\vec{i}-\vec{j})+(\vec{i}+4\vec{j})\).

10. Vectors with a given length

Complete each part.

1. Find the vector of length 20 that points in the same direction as \(\langle 3, 4\rangle\).
2. Find the vector of length 20 that points in the opposite direction.
3. Find the vector of length 26 in the direction of \(\langle -5, 12\rangle\).

Precalculus Vectors Worksheet: problems 11–15

11. Parallelogram rule

The figure shows \(\vec{u}=\langle 4, 1\rangle\) and \(\vec{v}=\langle 1, 3\rangle\). Complete the parallelogram, then find the diagonal \(\vec{u}+\vec{v}\), the vector \(\vec{u}-\vec{v}\), and the length of the diagonal to the nearest hundredth.

12. Tip-to-tail path

A drone flies three legs, shown below: \(\vec{a}=\langle 2, 3\rangle\), \(\vec{b}=\langle 4, -1\rangle\) and \(\vec{c}=\langle -5, 2\rangle\), with distances in blocks. Find the single vector that takes the drone from its start to its end. Then find the magnitude and direction angle of that vector.

13. Dot products

Find each dot product, then state which pair of vectors is perpendicular.

1. \(\langle 2, 3\rangle\cdot\langle 4, -1\rangle\)
2. \(\langle 3, -4\rangle\cdot\langle 4, 3\rangle\)
3. \(\langle -2, 6\rangle\cdot\langle 1, 2\rangle\)
4. \(\langle -3, -5\rangle\cdot\langle 2, -4\rangle\)

14. Angle between two vectors

Find the angle between each pair of vectors to the nearest tenth of a degree.

1. \(\langle 1, 0\rangle\) and \(\langle 1, 1\rangle\)
2. \(\langle 3, 1\rangle\) and \(\langle 1, 3\rangle\)
3. \(\langle 2, -1\rangle\) and \(\langle -4, 2\rangle\)
4. \(\langle -1, 2\rangle\) and \(\langle 3, 1\rangle\)

15. Perpendicular and parallel vectors

Complete each part.

1. Find \(k\) so that \(\langle k, 3\rangle\) is perpendicular to \(\langle 4, -6\rangle\).
2. Find \(k\) so that \(\langle k, 6\rangle\) is parallel to \(\langle 2, 3\rangle\).
3. Find a vector of length \(\sqrt{29}\) that is perpendicular to \(\langle 5, -2\rangle\).

Precalculus Vectors Worksheet: problems 16–20

16. Angles of a triangle

Triangle \(ABC\) has vertices \(A(0, 0)\), \(B(4, 0)\) and \(C(1, 3)\). Use vectors and the dot product to find all three interior angles to the nearest hundredth of a degree, and check that they add to \(180^{\circ}\).

17. Hiking displacement

Maya hikes 3 miles east, then 4 miles north, then 6 miles west. Write her total displacement as a vector (east and north positive), then find how far she is from the trailhead and the direction angle of the straight line back from the trailhead to her.

18. Boat crossing a river

A boat crosses a river 0.6 mile wide, heading straight across at 12 mph in still water. The current flows downstream at 5 mph. Find the boat’s speed relative to the ground, the time to cross, and how far downstream the boat lands from the point directly across.

19. Airplane and crosswind

A plane flies due north with an airspeed of 300 mph. A wind blows due east at 50 mph. Find the plane’s ground speed and its direction measured east of north. How far does it travel over the ground in 2 hours?

20. Forces in equilibrium

Two ropes pull on a ring with a force of 30 pounds due east and 40 pounds due north. Find the resultant force and its direction angle. Then find the third force that would hold the ring in equilibrium (net force zero).

Precalculus Vectors Worksheet: problems 21–22

21. Work done by a force

Diego pulls a sled with a constant force of 40 pounds at an angle of \(60^{\circ}\) above the horizontal. The sled moves 25 feet along level ground. Write the force and displacement as vectors and use the dot product to find the work done.

22. Crate on a ramp

A 100-pound crate rests on a ramp inclined at \(30^{\circ}\). Its weight is the vector \(\langle 0, -100\rangle\). The unit vector pointing down the ramp is \(\langle \cos 30^{\circ},\ -\sin 30^{\circ}\rangle\). Use the dot product to find the part of the weight that pulls the crate down the ramp, and the force needed to hold it in place.

Which skills do problems 1 to 10 build?

Problem 1 turns points \(A(2,1)\) and \(B(7,13)\) into \(\langle5,12\rangle\), whose magnitude is 13. Next, problem 3 asks for direction angles between \(0^{\circ}\) and \(360^{\circ}\), so quadrants matter. Problem 8 divides \(\langle8,-6\rangle\) by 10 to build a unit vector.

How do problems 13 to 16 use the dot product?

Problem 15 asks for \(k\) so that \(\langle k,3\rangle\) is perpendicular to \(\langle4,-6\rangle\). Set \(4k-18=0\), which gives \(k=4.5\). Problem 16 then finds the angles of triangle \(ABC\) with \(A(0,0)\), \(B(4,0)\) and \(C(1,3)\), and the angle at \(B\) is \(45^{\circ}\).

What does a precalculus vectors worksheet teach through applications?

Problem 17 adds three hiking legs into one displacement of \(\langle-3,4\rangle\), which has length 5 miles. In problem 19, a 300 mph plane with a 50 mph crosswind has a ground speed near 304 mph. Problem 22 resolves a 100-pound weight on a \(30^{\circ}\) ramp.

Which errors appear most often?

• Subtracting the start point from the end point in reverse order.
• Using \(\tan^{-1}\) without checking the quadrant.
• Treating the dot product as a vector.

How can you spread these 22 vector problems over two days?

Use day one for components, unit vectors and sums. Meanwhile, keep the dot product and the applications for day two. Sketch every vector before you compute anything, and compare each sketch with your final answer.

See every chapter on the Precalculus math page.