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C. Geometry and Algebra of Vectors · Dot and Cross Products

Dot and Cross Products: Unit Review

C2.4 C2.5 C2.6 C2.7 C2.8About 40 minutes
Learning goals
  • I can compute and apply dot products: angles, orthogonality, projections and work.
  • I can compute and apply cross products: perpendicular vectors, areas, parallel tests and torque.
  • I can use the triple scalar product for volumes and coplanarity, and choose the right tool for a problem.

Words to know

dot product
u · v = |u||v| cos θ = u₁v₁ + u₂v₂ + u₃v₃, a scalar.
cross product
u × v = [u₂v₃ − u₃v₂, u₃v₁ − u₁v₃, u₁v₂ − u₂v₁], a vector perpendicular to u and v with length |u||v| sin θ.
triple scalar product
a · (b × c); its absolute value is the volume of the parallelepiped on a, b and c.

Let's review!

In this unit you met two new ways to multiply vectors. The dot product u · v = |u||v| cos θ = u₁v₁ + u₂v₂ + u₃v₃ gives a number. You used it to find angles, test for right angles, find projections and compute work. The cross product u × v gives a vector perpendicular to both, with length |u||v| sin θ. You used it to find normals, areas, torque, and, combined with the dot product, volumes and coplanarity.

The next unit, Lines and Planes, uses these tools constantly: dot products to test perpendicular lines and planes, cross products to find normal vectors and direction vectors. This review has a few questions from every lesson. Try each one, then press Show answer.

How to use this review

Do one or two sections at a time. Write every vector in components before you start, and keep exact values (fractions and radicals) until the last step. After every cross product, dot your answer with both original vectors to check for 0. Use degree mode. For each question you miss, write down why (wrong formula? sign slip? wrong order?) and go back to that lesson. Finish with the 10-question quiz.

Dot product u · vresult: a number= |u||v| cos θ= u₁v₁ + u₂v₂ + u₃v₃equals 0 ⇔ perpendicularangles, projections, workCross product u × vresult: a vector (3-D only)length |u||v| sin θ = areaperpendicular to u and vequals 0 ⇔ parallelnormals, areas, torque
The two products side by side. The dot product gives a number built from cos θ; the cross product gives a perpendicular vector whose length, built from sin θ, is an area.
Just for fun

Dot product or cross product? If you want a number, connect the dots. If you want a new direction, get cross.

The Dot Product: Geometric Definition

1

|u| = 8, |v| = 3 and θ = 60°. Find u · v.

2

|a| = 5, |b| = 2 and θ = 135°. Find a · b exactly.

3

u · v = −20, |u| = 5 and |v| = 8. Find θ.

The Dot Product in Cartesian Form

1

Find [4, −1] · [2, 7].

2

Find [3, 0, −2] · [1, 5, 4].

3

Find k if [k, 2, 1] · [3, k, −4] = 6.

Properties of the Dot Product and Orthogonal Vectors

1

|a| = 2, |b| = 3 and a · b = −1. Find |a + b|.

2

Find k so that [2, k, 3] and [k, 4, −2] are orthogonal.

3

Expand (a − b) · (a + 2b).

Angles Between Vectors and Direction Angles

1

Find the angle between [1, 1, 0] and [0, 1, 1].

2

Find the direction angles of [1, 2, 2].

3

A(1, 1), B(4, 2) and C(3, 5) are vertices of a triangle. Is it a right triangle?

Scalar and Vector Projections

1

Find the scalar and vector projections of [4, 2] on [3, −4].

2

Find the scalar and vector projections of [1, −1, 2] on [2, 1, 2].

3

Write [6, 2] as a vector parallel to [1, 1] plus a vector perpendicular to it.

Work Done by a Force

1

A 30 N force at 20° to the displacement moves a box 10 m. Find the work.

2

F = [5, −2, 3] N moves an object from (1, 1, 1) to (3, 4, 0) (metres). Find the work.

3

Why does the upward push of a level road do no work on a car driving along it?

What do you need?angle, right-angle test,projection, workdot productperpendicular vector,area, parallel test, torquecross productvolume, or are threevectors coplanar?triple scalar product
Choosing a tool: start from what the question asks for, then pick the product that delivers it.

The Cross Product

1

Find [1, −1, 2] × [3, 0, 1] and check it.

2

Find k × j.

3

Find both unit vectors perpendicular to [1, 0, 0] and [0, 1, 1].

Magnitude of the Cross Product: Areas of Parallelograms and Triangles

1

Find the area of the parallelogram determined by [1, −1, 2] and [3, 0, 1].

2

Find the area of the triangle with vertices P(0, 0, 0), Q(1, 2, 0) and R(2, 0, 2).

3

|u| = 3, |v| = 10 and θ = 30°. Find |u × v|.

Properties of the Cross Product

1

a × b = [1, −2, 5]. Find (−2b) × a.

2

Simplify (a + 2b) × (3a − b).

3

True or false: (a × b) × c = a × (b × c) for all vectors.

Torque

1

A 50 N force acts at 30° to a 0.4 m wrench. Find the torque.

2

F = [0, 0, 30] N acts at r = [0.2, 0.1, 0] m. Find τ and |τ|.

3

Why are door handles placed far from the hinges?

The Triple Scalar Product: Volume and Coplanar Vectors

1

Find the volume of the parallelepiped determined by [1, 0, 1], [0, 2, 0] and [1, 1, 3].

2

Are [1, 2, 3], [2, 1, 0] and [3, 3, 3] coplanar?

3

Find k so that [1, k, 0], [0, 1, 1] and [1, 0, 1] are coplanar.

Solving Problems with Dot and Cross Products

1

A plane has air velocity [0, 250] km/h and the wind is [−40, 0] km/h. Find the ground speed and the angle between heading and track.

2

The wind is [6, −8] km/h and a runway points north, [0, 1]. Find the headwind and crosswind for a plane taking off northward.

3

You know three edge vectors of a crystal. Which tool gives its volume?

Check your understanding

Quiz

1.

|u| = 4, |v| = 5 and θ = 60°. What is u · v?

2.

What is [2, −3, 1] · [4, 1, −5]?

3.

What is the angle between [1, 0, 1] and [1, 1, 0]?

4.

What is the vector projection of [3, 4] on [1, 0]?

5.

A force F = [2, 0, 1] N moves an object through d = [5, 3, 4] m. How much work is done?

6.

What is [0, 1, 0] × [0, 0, 1]?

7.

What is the area of the triangle with sides [2, 0, 0] and [0, 4, 0] from one vertex?

8.

u × v = [1, −1, 2]. What is v × u?

9.

A 100 N force is applied perpendicular to a 0.3 m wrench. What is the torque?

10.

What is the volume of the parallelepiped determined by [1, 0, 0], [0, 1, 0] and [1, 1, 5]?

For parents and teachers

Use the review to find what to revisit:

  • Using the inside angle of a triangle, or wrong signs for obtuse angles → Lesson 1 (The Dot Product: Geometric Definition).
  • Arithmetic of "multiply matching components and add" → Lesson 2 (The Dot Product in Cartesian Form).
  • Expanding |a + b|², or finding orthogonal vectors → Lesson 3 (Properties of the Dot Product and Orthogonal Vectors).
  • Angles in triangles or direction angles → Lesson 4 (Angles Between Vectors and Direction Angles).
  • Dividing by the wrong length, or mixing scalar and vector projections → Lesson 5 (Scalar and Vector Projections).
  • Work problems, signs of work → Lesson 6 (Work Done by a Force).
  • Sign slips in the cross product formula, right-hand rule → Lesson 7 (The Cross Product).
  • Areas, forgetting the ½ for triangles, parallel tests → Lesson 8 (Magnitude of the Cross Product).
  • Changing the order of a cross product without changing the sign → Lesson 9 (Properties of the Cross Product).
  • Torque, or using cos instead of sin → Lesson 10 (Torque).
  • Volumes and coplanarity → Lesson 11 (The Triple Scalar Product).
  • Choosing a tool, heading and track, headwind and crosswind → Lesson 12 (Solving Problems with Dot and Cross Products).

Looking ahead: the next unit, Lines and Planes, describes lines and planes in two-space and three-space with equations. Normal vectors (cross products), perpendicularity tests (dot products) and distances (projections and cross products) appear in almost every lesson. In university, these ideas lead to linear algebra, multivariable calculus, physics and engineering mechanics, and computer graphics.