Vector Product

  2. Geometry & Trigonometry
  3. Vector Product
    4. 30'

The vector product is a way of combining two vectors to find a third vector that is perpendicular to the first two. The vector product is a vector! Don't mix it up with the  scalar product which is a scalar! The vector product is useful for finding the equation of a plane. The properties are important, so learn them. The two following formula give the definition of the vector product

\(\textbf{v}\times \textbf{w}=\left( \begin{matrix} { \textbf{v} }_{ 1 } \\ { \textbf{v} }_{ 2 } \\ { \textbf{v} }_{ 3 } \end{matrix} \right) \times \left( \begin{matrix} { \textbf{w} }_{ 1 } \\ { \textbf{w} }_{ 2 } \\ { \textbf{w} }_{ 3 } \end{matrix} \right) =\left( \begin{matrix} { \textbf{v} }_{ 2 }{ \textbf{w} }_{ 3 }-{ \textbf{v} }_{ 3 }{ \textbf{w} }_{ 2 } \\ { \textbf{v} }_{ 3 }{ \textbf{w} }_{ 1 }-{ \textbf{v} }_{ 1 }{ \textbf{w} }_{ 3 } \\ { \textbf{v} }_{ 1 }{ \textbf{w} }_{ 2 }-{ \textbf{v} }_{ 2 }{ \textbf{w} }_{ 1 } \end{matrix} \right)\)

\(\textbf{v}\times \textbf{w}=\left| \textbf{v} \right| \left| \textbf{w} \right| \sin\theta \ \textbf{n}\)

where \(\theta\) is the angle between v and w and n is the unit normal vector

Key Concepts

On this page, you should learn about

  • the vector product (or cross product) of two vectors
  • properties of the vector product
  • the magnitude of the vector product to find the area of a parallelogram
 

Essentials

The following videos will help you understand all the concepts from this page

Vector Product - How to

In the following video we see how to work out the vector product using the formula, what this means and how to use the right-hand rule to find its direction.

​​ Notes from the video

 

Properties of Vector Product

There are 4 important properties of the vector product that you will need to learn. These properties are often used in proofs.

\(\textbf{v}\times\textbf{w} =-\textbf{w} \times \textbf{v}\\ \textbf{u} \times (\textbf{v}+\textbf{w} )= \textbf{u} \times \textbf{v}+\textbf{u} \times \textbf{w} \\ (k\textbf{v})\times \textbf{w} =k(\textbf{v}\times \textbf{w})\\ \textbf{v}\times \textbf{v}=0 \)


Here is an example of a proof involving the vector product

Show that \((\textbf{v}+\textbf{w})\times (\textbf{v}-\textbf{w})=2\textbf{w}\times \textbf{v}\)

​​ Notes from the video

 

Magnitude of Vector Product

We know that \(\textbf{v}\times \textbf{w}= \color{red} {\left| \textbf{v} \right| \left| \textbf{w} \right| sin\theta} \ \textbf{n}\)

where \(\theta\) is the angle between v and w and n is the unit normal vector.

Hence if we find the magnitude of the vector product

\(\color{red} {\left|\textbf{v}\times \textbf{w}\right|} = \color{red} {\left| \textbf{v} \right| \left| \textbf{w} \right| sin\theta} \)

If we divide this by two, we get something familiar

\(\frac {1}{2} {\left|\textbf{v}\times \textbf{w}\right|} = \frac {1}{2} {\left| \textbf{v} \right| \left| \textbf{w} \right| sin\theta} \)

You may recall that formula for the area of a triangle is \(A = \frac {1}{2}absin \theta\)

Hence...

Area of a triangle = \(\frac {1}{2} {\left|\textbf{v}\times \textbf{w}\right|}\)

It therefore follows that for a paralellogram...

Area of a parallelogram = \(\left| \textbf{v}\times \textbf{w} \right| \)

In the following video, we are going to look at this application of the vector product. In this example we use the vector product to find the area of a parallelogram:

A parallelogram has two adjacent sides formed by the vectors 2i - j + 3k and ai + 4k. The area of the parallelogram is \(\sqrt{26}\). Find the possible values of a.

​​ Notes from the video

 

Vector and Scalar Product

Here is an example that combines the vector product and the scalar product

Prove that \({ \left| \textbf{v}\times \textbf{w} \right| }^{ 2 }={ \left| \textbf{v} \right| }^{ 2 }{ \left| \textbf{w} \right| }^{ 2 }-{ (\textbf{v}\cdot \textbf{w}) }^{ 2 }\)

​​ Notes from the video

 
 

Summary

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Test Yourself

Here is a quiz that practises the skills from this page


START QUIZ!

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Exam-style Questions

Question 1

a = \(\left( \begin{matrix} 2 \\ 3 \\ -5 \end{matrix} \right) \) and b = \(\left( \begin{matrix} 3 \\ -2 \\ 4 \end{matrix} \right) \)

Find \(\textbf{a}\times \textbf{b}\)

Hint

Full Solution

 

 

Question 2

a = 3i + 2j + k and b = 2i + 3j + 2k

Find a unit vector that is perpendicular to a and b

Hint

Full Solution

 

 

Question 3

The area of a parallelogram formed by two adjacent vectors a and b is 7 square units.

a = \(\left( \begin{matrix} -3 \\ 4 \\ k \end{matrix} \right) \) b = \(\left( \begin{matrix} 3 \\ -2 \\ -2 \end{matrix} \right) \)

Find k

Hint

Full Solution

 

 

Question 4

Given that \(\textbf{u}\times \textbf{v} = \textbf{u}\times \textbf{w}\) show that \(\textbf{v}- \textbf{w} \) is parallel to \(\textbf{u}\)

Hint

Full Solution

 

 

Question 5

a) For any two vectors v and w prove Lagrange's Identity

\({ \left| \textbf{v}\times \textbf{w} \right| }^{ 2 }+{ \left( \textbf{v}\cdot \textbf{w} \right) }^{ 2 }={ \left| \textbf{v} \right| }^{ 2 }{ \left| \textbf{w} \right| }^{ 2 }\)

b) Hence, find \(\textbf{v}\cdot \textbf{w}\) if

\({ \left| \textbf{v} \right| }=3\)

\({ \left| \textbf{w} \right| }=4\)

\(\textbf{v}\times \textbf{w} =\left( \begin{matrix} -1 \\ 2 \\ 3 \end{matrix} \right) \)

Hint

Full Solution

 

 

Question 6

The points A, B and C are given by the position vectors a, b and c.

If A, B and C are collinear, show that

\(\textbf{b}\times \textbf{c}=\textbf{a}\times (\textbf{c}-\textbf{b})\)

Hint

Full Solution

 

 

Question 7

a and b are vectors

Show that \(|\textbf{a}×\textbf{b}|^{ 2 }+|\textbf{a} ∙\textbf{b}|^{ 2 }=(\textbf{a}\textbf{b})^{ 2 }\)

Hint

Full Solution

 
MY PROGRESS

How much of Vector Product have you understood?