\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \)
\( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \)
17calculus
Limits Derivatives Integrals Infinite Series Parametrics Polar Coordinates Conics
Limits
Epsilon-Delta Definition
Finite Limits
One-Sided Limits
Infinite Limits
Trig Limits
Pinching Theorem
Indeterminate Forms
L'Hopitals Rule
Limits That Do Not Exist
Continuity & Discontinuities
Intermediate Value Theorem
Derivatives
Power Rule
Product Rule
Quotient Rule
Chain Rule
Trig and Inverse Trig
Implicit Differentiation
Exponentials & Logarithms
Logarithmic Differentiation
Hyperbolic Functions
Higher Order Derivatives
Differentials
Slope, Tangent, Normal...
Linear Motion
Mean Value Theorem
Graphing
1st Deriv, Critical Points
2nd Deriv, Inflection Points
Related Rates Basics
Related Rates Areas
Related Rates Distances
Related Rates Volumes
Optimization
Integrals
Definite Integrals
Integration by Substitution
Integration By Parts
Partial Fractions
Improper Integrals
Basic Trig Integration
Sine/Cosine Integration
Secant/Tangent Integration
Trig Integration Practice
Trig Substitution
Linear Motion
Area Under/Between Curves
Volume of Revolution
Arc Length
Surface Area
Work
Moments, Center of Mass
Exponential Growth/Decay
Laplace Transforms
Describing Plane Regions
Infinite Series
Divergence (nth-Term) Test
p-Series
Geometric Series
Alternating Series
Telescoping Series
Ratio Test
Limit Comparison Test
Direct Comparison Test
Integral Test
Root Test
Absolute Convergence
Conditional Convergence
Power Series
Taylor/Maclaurin Series
Radius of Convergence
Interval of Convergence
Remainder & Error Bounds
Fourier Series
Study Techniques
Choosing A Test
Sequences
Infinite Series Table
Practice Problems
Exam Preparation
Exam List
Parametrics
Parametric Curves
Parametric Surfaces
Slope & Tangent Lines
Area
Arc Length
Surface Area
Volume
Polar Coordinates
Converting
Slope & Tangent Lines
Area
Arc Length
Surface Area
Conics
Parabolas
Ellipses
Hyperbolas
Conics in Polar Form
Vectors Vector Functions Partial Derivatives/Integrals Vector Fields Laplace Transforms Tools
Vectors
Unit Vectors
Dot Product
Cross Product
Lines In 3-Space
Planes In 3-Space
Lines & Planes Applications
Angle Between Vectors
Direction Cosines/Angles
Vector Projections
Work
Triple Scalar Product
Triple Vector Product
Vector Functions
Projectile Motion
Unit Tangent Vector
Principal Unit Normal Vector
Acceleration Vector
Arc Length
Arc Length Parameter
Curvature
Vector Functions Equations
MVC Practice Exam A1
Partial Derivatives
Gradients
Directional Derivatives
Lagrange Multipliers
Tangent Plane
MVC Practice Exam A2
Partial Integrals
Describing Plane Regions
Double Integrals-Rectangular
Double Integrals-Applications
Double Integrals-Polar
Triple Integrals-Rectangular
Triple Integrals-Cylindrical
Triple Integrals-Spherical
MVC Practice Exam A3
Vector Fields
Curl
Divergence
Conservative Vector Fields
Potential Functions
Parametric Curves
Line Integrals
Green's Theorem
Parametric Surfaces
Surface Integrals
Stokes' Theorem
Divergence Theorem
MVC Practice Exam A4
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Prepare For Calculus 1
Ready For Calculus 2?
Trig Formulas
Describing Plane Regions
Parametric Curves
Linear Algebra Review
Word Problems
Mathematical Logic
Calculus Notation
Simplifying
Practice Exams
17calculus on YouTube
More Math Help
Tutoring
Tools and Resources
Academic Integrity
Learning/Study Techniques
Math/Science Learning
Memorize To Learn
Music and Learning
Note-Taking
Motivation
Instructor or Coach?
Books
Math Books
How To Read Math Books

You CAN Ace Calculus

17calculus > vectors > cross product

Vector Cross Product

on this page: ► calculating the cross product     ► cross product properties     ► cross product applications

In order to understand the material on this page, you need to know some linear algebra, specifically, how to calculate the determinant of 2x2 and 3x3 matrices. You can find a quick review on the linear algebra page.

Calculating The Cross Product

The Cross Product is one way to 'multiply' two vectors (the other way is the dot product). Unlike the dot product, the cross product only makes sense when performed on two 3-dim vectors. Taking the cross product of the two vectors \( 3\hat{i}+2\hat{j} \) and \( \hat{i}+\hat{j} \) is not possible, unless you mean \( 3\hat{i}+2\hat{j}+0\hat{k} \) and \( \hat{i}+\hat{j}+0\hat{k} \), in which case, you need to write out the \(\hat{k}\) term even if it is zero.

If we have two vectors, \(\vec{u}=\langle u_1, u_2, u_3 \rangle \) and \(\vec{v} = \langle v_1, v_2, v_3 \rangle \), we write the cross product of these two vectors as \( \vec{u} \times \vec{v} \).

The result of the cross product of two vectors is another vector. It's meaning is discussed later on this page. For now, let's focus on how we calculate the cross product.

To calculate the cross product, we use some linear algebra. If you haven't already, now would be good time to review the linear algebra page to make sure your skills calculating a 3x3 determinant are sharp. To calculate the cross product we calculate the following determinant.

\( \vec{u} \times \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix} \) \( = (u_2 v_3 - u_3 v_2)\hat{i} - (u_1 v_3 - u_3 v_1)\hat{j} + (u_1 v_2 - u_2 v_1)\hat{k} \)

Comments -
1. It is best NOT to memorize the last expression. Instead, set up and evaluate the determinant.
2. Remember to subtract the middle term.
3. It is important to set up the determinant correctly, i.e.
- The first row is the set of unit vectors.
- The second row is the first vector of the cross product.
- The third row is the second vector of the cross product.
The rows cannot be in any other order (more on this in the properties section below).

The name 'cross product' comes from the notation using '\(\times\)' between the two vectors. Just like with the dot product, it is important to use the '\(\times\)' between the vectors to indicate a cross product. Writing \(\vec{u} \vec{v} \) makes no sense and is considered incorrect notation.

Note: Recently we heard that what we call the 'determinant' above is not strictly a determinant but just a mnemonic device to calculate the cross product, since a true determinant consists of only numbers not vectors. We have not verified this at this time but it certainly could be true. When we verify it, we will update this page accordingly.

Okay, so let's watch a video clip discussing the cross product and its geometric interpretation.

PatrickJMT - geometric interpretation of the cross product

Cross Product Properties

Here are some cross product properties.
Algebraic Properties
Let \(\vec{u}\), \(\vec{v}\) and \(\vec{w}\) be vectors in space and let \(a\) be a scalar.
1. \(\vec{u} \times \vec{v} = -(\vec{v} \times \vec{u})\)
2. \(\vec{u} \times (\vec{v}+\vec{w}) = (\vec{u} \times \vec{v}) + (\vec{u} \times \vec{w}) \)
3. \(a(\vec{u} \times \vec{v}) = (a\vec{u}) \times \vec{v} = \vec{u} \times (a\vec{v})\)
4. \(\vec{u} \times \vec{0} = \vec{0} \)
5. \( \vec{u} \times \vec{u} = \vec{0} \)
Notes:
- property 1 implies that the cross product is not commutative
- in property 2, the vector \(\vec{u}\) is on the left, so when it is distributed across the addition, it must remain on the left in both cases
- property 5 seems trivial but it is very powerful; later on this page, this will be discussed in more detail.

Here is a video with proofs of some of these algebraic properties.

Larson Calculus - proofs of some algebraic properties

Geometric Properties
Let \(\vec{u}\) and \(\vec{v}\) be nonzero vectors in space and let \(\theta\) be the angle between \(\vec{u}\) and \(\vec{v}\).
6. \(\vec{u} \times \vec{v}\) is orthogonal to both \(\vec{u}\) and \(\vec{v}\).
7. \( \| \vec{u} \times \vec{v} \| = \|\vec{u}\| \|\vec{v}\| \sin \theta\)
8. \(\vec{u} \times \vec{v} = \vec{0} \) if and only if \(\vec{u}\) is a scalar multiple of \(\vec{v}\)
9. \( \| \vec{u} \times \vec{v} \| \) represents the area of the parallelogram formed with \(\vec{u}\) and \(\vec{v}\) as adjacent sides.
Notes:
- for the geometric properties, both vectors must be nonzero; this is not a requirement in the algebraic properties
- notice in property 7, the cross product involves the sine of angle \(\theta\) while the dot product involves the cosine of the angle
- as mentioned in property 9, the cross product is the area of a parallelogram; here is a great video that discusses this in more detail.

Dr Chris Tisdell - Cross Product and Area of Parallelogram

Here is a video with proofs of some of these geometric properties.

Larson Calculus - proofs

There is a simple rule to use when you need to know the direction of the resulting vector from the cross product. It's called the right hand rule. The idea is to lay out your hand with all fingers straight out. Place the middle of your hand at the point of intersection of the two vectors involved in the cross product with your fingers in the direction of the first vector. Fold your fingers in the direction of the second vector. Your thumb will then be pointing in the direction of the result of the cross product. Here is a quick video showing this idea.

Right Hand Rule for Cross Products

All this information may be a bit overwhelming. So let's take a few minutes and watch this video. He explains the cross product very well and shows some examples.

Dr Chris Tisdell - Cross product of vectors

Cross Product Applications

Triple Scalar Product

Triple Scalar Product

The triple scalar product is a result of combining the dot product with the cross product. First, let's define what it is and then discuss a couple of properties.

Definition and Notation - If we have three vectors in space, \(\vec{u} = u_x\hat{i}+u_y\hat{j}+u_z\hat{k}\), \(\vec{v} = v_x\hat{i}+v_y\hat{j}+v_z\hat{k}\) and \(\vec{w} = w_x\hat{i}+w_y\hat{j}+w_z\hat{k}\), then the triple scalar product is defined to be \( \vec{u} \cdot (\vec{v} \times \vec{w}) \)
The calculation of this can be done as follows
\( \vec{u} \cdot (\vec{v} \times \vec{w}) = \begin{vmatrix} u_x & u_y & u_z \\ v_x & v_y & v_z \\ w_x & w_y & w_z \end{vmatrix} \)     [ proof ]

The triple scalar product is so named because the result is a scalar. [For comparison, see the triple vector product panel below.]

Properties - The triple scalar product can also be evaluated in one of the following forms.
\( \vec{a} \cdot (\vec{b} \times \vec{c}) = \vec{b} \cdot (\vec{c} \times \vec{a}) = \vec{c} \cdot (\vec{a} \times \vec{b}) \)
The parentheses may be omitted since evaluating the dot product first yields a scalar and it doesn't make sense to take the cross product of a scalar with a vector.
This property also holds \( [\vec{a} \cdot (\vec{b} \times \vec{c})]\vec{a} = (\vec{a} \times \vec{b}) \times (\vec{a} \times \vec{c}) \)
When the triple scalar product is zero, the 3 vectors are coplanar.

Applications - Geometrically, the triple scalar product \( \vec{a} \cdot (\vec{b} \times \vec{c} ) \) is the (signed) volume of the parallelepiped defined by the three vectors given (see figure on the right). The word 'signed' means that the result can be positive or negative depending on the orientation of the vectors.
You can probably now see that when the 3 vectors are coplanar, the parallelepiped is flat and has no volume, so the triple scalar product is zero.

Other Names For the Triple Scalar Product -
- scalar triple product
- mixed product
- box product

Here are several videos that explains this in more detail. The first two are especially good and the third contains a proof.

Dr Chris Tisdell - Scalar triple product

Dr Chris Tisdell - Scalar triple product and volume

Larson Calculus -Proof - Geometric Property of the Triple Scalar Product

Triple Vector Product

Triple Vector Product

The triple vector product (or vector triple product, as it is sometimes called) is so named because the result is a vector. [For comparison, see the triple scalar product panel above.]

When you have three vectors, \( \vec{u}\), \(\vec{v}\) and \(\vec{w}\), the triple vector product is defined as \( \vec{u} \times \vec{v} \times \vec{w} \).

Triple Scalar Product Proof

Theorem: Triple Scalar Product

If we have three vectors in space,
\(\vec{u} = u_x\hat{i}+u_y\hat{j}+u_z\hat{k}\), \(\vec{v} = v_x\hat{i}+v_y\hat{j}+v_z\hat{k}\) and \(\vec{w} = w_x\hat{i}+w_y\hat{j}+w_z\hat{k}\),
then the triple scalar product is \( \vec{u} \cdot (\vec{v} \times \vec{w}) = \begin{vmatrix} u_x & u_y & u_z \\ v_x & v_y & v_z \\ w_x & w_y & w_z \end{vmatrix} \)

Proof - - To prove this, we will calculate \( \vec{u} \cdot (\vec{v} \times \vec{w}) \) and then calculate the determinant to show that we get the same result. Let's start with the cross product.

\( \vec{v} \times \vec{w} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ v_x & v_y & v_z \\ w_x & w_y & w_z \end{vmatrix} = \) \( (v_y w_z - v_z w_y)\hat{i} - \) \( (v_x w_z - w_x v_z)\hat{j} + \) \( (v_x w_y - w_x v_y)\hat{k} \)

Taking the dot product of \(\vec{u}\) and the last equation gives us

\( (u_x\hat{i}+u_y\hat{j}+u_z\hat{k}) \cdot [ (v_y w_z - v_z w_y)\hat{i} - \) \( (v_x w_z - w_x v_z)\hat{j} + \) \( (v_x w_y - w_x v_y)\hat{k} ] = \) \( u_x(v_y w_z - v_z w_y) - \) \( u_y(v_x w_z - w_x v_z) + \) \( u_z(v_x w_y - w_x v_y) \)

We could certainly multiply the components of vector \(\vec{u}\) through each factor, but for reasons you will see later, we will leave the equation as it is. To sum up, we have calculated

\( \vec{u} \cdot (\vec{v} \times \vec{w}) = \) \( u_x(v_y w_z - v_z w_y) - \) \( u_y(v_x w_z - w_x v_z) + \) \( u_z(v_x w_y - w_x v_y) ~~~~~ (1) \)

Okay, now let's calculate the determinant
\( \begin{vmatrix} u_x & u_y & u_z \\ v_x & v_y & v_z \\ w_x & w_y & w_z \end{vmatrix} \)

We will go across the top row (although going down the first column will give us the same result, going across the top row makes the algebra come out the way want it to).

\( \begin{vmatrix} u_x & u_y & u_z \\ v_x & v_y & v_z \\ w_x & w_y & w_z \end{vmatrix} = \) \( u_x(v_y w_z - v_z w_y) - \) \( u_y(v_x w_z - w_x v_z) + \) \( u_z(v_x w_y - w_x v_y) \)

Notice this last equation is the same as equation (1) above. This is easier to see since we did not multiply out all factors in Equation (1). So, we have shown

\( \vec{u} \cdot (\vec{v} \times \vec{w}) = \begin{vmatrix} u_x & u_y & u_z \\ v_x & v_y & v_z \\ w_x & w_y & w_z \end{vmatrix} \)       [qed]

Search 17Calculus

Practice Problems

Instructions - - Unless otherwise stated, give all your answers in exact form. For angles, give your answers in radians to 3 decimal places.

Level A - Basic

Practice A01

Given \(\vec{A}=2\hat{i}+3\hat{j}+4\hat{k}\) and \(\vec{B}=\hat{i}+3\hat{k}\), calculate the cross product \(\vec{A}\times\vec{B}\).

answer

solution

Practice A02

Given \(\vec{A}=2\hat{i}+3\hat{j}+4\hat{k}\) and \(\vec{B}=\hat{i}+3\hat{k}\), use the cross product to find the angle between \(\vec{A}\) and \(\vec{B}\).

answer

solution

Practice A03

Find a unit vector that is perpendicular to \(\vec{A}=\hat{i}+2\hat{j}+3\hat{k}\) and \(\vec{B}=3\hat{i}+2\hat{j}+\hat{k}\).

answer

solution

Practice A04

Show that the vectors \(\vec{A} = 2\hat{i}-3\hat{j}+4\hat{k}\), \(\vec{B} = 6\hat{i}+2\hat{j}+\hat{k}\) and \(\vec{C} = 6\hat{i}+10\hat{j}-7\hat{k}\) are coplanar.

answer

solution

Practice A05

Show that the vectors \(\vec{A}=2\hat{i}+3\hat{j}+6\hat{k}\) and \(\vec{B}=6\hat{i}+2\hat{j}-3\hat{k}\) are perpendicular.

answer

solution

Practice A06

Find a unit vector that is perpendicular to \(\vec{A}=2\hat{i}+3\hat{j}+6\hat{k}\) and \(\vec{B}=6\hat{i}+2\hat{j}-3\hat{k}\).

answer

solution

Practice A07

Given \(\vec{A}=\hat{i}-2\hat{j}+2\hat{k}\) and \(\vec{B}=3\hat{i}-\hat{j}-\hat{k}\) calculate the cross product \(\vec{A}\times\vec{B}\).

answer

solution

Practice A08

Calculate the cross product of \(\hat{i}\) and \(\hat{j}\).

answer

solution

Practice A09

Given \(\vec{A}=\hat{i}-2\hat{j}+2\hat{k}\), \(\vec{B}=3\hat{i}-\hat{j}-\hat{k}\) and \(\vec{C}=-\hat{i}-\hat{k}\), calculate the triple scalar product \(\vec{A}\cdot\vec{B}\times\vec{C}\).

answer

solution

Practice A10

Given \(\vec{A}=\hat{i}-2\hat{j}+2\hat{k}\), \(\vec{B} = 3\hat{i}-\hat{j}-\hat{k}\) and \(\vec{C}=-\hat{i}-\hat{k}\), calculate the triple vector product \(\vec{A}\times\vec{B}\times\vec{C}\).

answer

solution

Practice A11

Calculate the cross product of the vectors \(\vec{a}=\langle5,-1,-2\rangle\) and \(\vec{b}=\langle-3,2,4\rangle\).

solution

Practice A12

Calculate the cross product of the vectors \(\vec{a}=\hat{i}-\hat{j}+3\hat{k}\) and \(\vec{b}=-2\hat{i}+3\hat{j}+\hat{k}\).

solution

Practice A13

Calculate the cross product of the vectors \(\vec{a}=\langle2,-3\rangle\) and \(\vec{b}=\langle4,5\rangle\).

solution

Practice A14

Calculate the cross product of the vectors \(\vec{a}=\langle5,1,4\rangle\) and \(\vec{b}=\langle-1,0,2\rangle\).

solution

Real Time Web Analytics
menu top search practice problems
17
menu top search practice problems 17