# Multiplying Vectors on TI-83/84

Copyright © 2005–2014 by Stan Brown, Oak Road Systems

Copyright © 2005–2014 by Stan Brown, Oak Road Systems

This page also gives a program, ready for downloading or
keying in, that computes those quantities plus the
**vector product (cross product)** of two vectors, the
**angle between** them, and the **area of the parallelogram**
that they form.

Your TI-83 or TI-84 can do operations on lists of numbers. Though it doesn’t know about vectors as such, you can consider a list of two or three numbers to be the components of a vector in the plane or in space.

Use **curly braces { }** around the list of
components. To get a left or right curly brace, press the
[`2nd`

] key and then the left or right parenthesis.

**Example:** Suppose vector **a** is [2,−3] and you want
to display the vector −7**a**. Here’s how.

Enter the scalar. | [`(-)` ] [`7` ] |

Enter the multiplication sign. | [`×` ] (displays as `*` ) |

Enter the vector in curly braces, with commas separating the components. | [`2nd` `(` makes `{` ] [`2` ] [`,` ] [`(-)` ] [`3` ] [`2nd` `)` makes `}` ] |

Display the result. |
[`ENTER` ] |

The dot product of two vectors **u** and **v** is formed
by multiplying their components and adding. In the plane,
**u·v** =
u_{1}v_{1}+u_{2}v_{2}; in space it’s
u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}.

If you tell the TI-83/84 to multiply two lists, it multiplies the elements of the two lists to make a third list. The sum of the elements of that third list is the dot product of the vectors.

Example: If **u** = [2,3,1] and **v** =
[4,−3,2], find **u·v**.

First invoke the summation function. | Press [`2nd` `STAT` makes `LIST` ] [`◄` ] [`5` ]
to make “sum(” appear. |

Enter the first vector, using curly braces as before. | [`2nd` `(` makes `{` ] [`2` ] [`,` ] [`3` ] [`,` ] [`1` ] [`2nd` `)` makes `}` ] |

Multiply. | Press [`×` ] and `*` appears on the
screen. |

Enter the second vector. | [`2nd` `(` makes `{` ] [`4` ] [`,` ] [`(-)` ] [`3` ] [`,` ] [`2` ] [`2nd` `)` makes `}` ] |

Close the parenthesis from “sum(” and do the
calculation. (u·v =
2×4 + 3×(−3) + 1×2 =
8−9+2 = 1.) |
[`)` ] [`ENTER` ] |

You know that the length or magnitude of vector **v** is
found by

||**v**|| = √(v_{1}²+v_{2}²)

You know also that the dot product of a vector with itself is

**v·v** =
v_{1}v_{1}+v_{2}v_{2} =
v_{1}²+v_{2}²

which is the square of the length of **v**. Therefore

||**v**|| = √(**v·v**)

With the TI-83/84, there’s no need to enter the vector twice to find its length. Instead, just square the list of components — the calculator interprets this as squaring every component. Then take the square root of the sum.

Example: Find the length of vector **a** =
[2,−5,−3].

Set up the square root. | Press [`2nd` `x²` makes `√` ]. Notice that the calculator
supplies a left parenthesis for you. |

Set up the sum. | [`2nd` `STAT` makes `LIST` ] [`◄` ] [`5` ] |

Enter the vector. | [`2nd` `(` makes `{` ] [`2` ] [`,` ] [`(-)` ] [`5` ] [`,` ] [`(-)` ] [`3` ] [`2nd` `)` makes `}` ] |

Square all the components and close both parentheses.
You can check the calculator’s result by computing the length manually: || v|| =
√2²+(−5)²+(−3)²) = 4+25+9 =
√38, which is about 6.16. |
[`x²` ] [`)` ] [`)` ] [`ENTER` ] |

I wrote a little TI-83/84 program to compute interesting results from two vectors, up to and including the cross product or vector product, and I offer it on this Web page.

There are three methods to get the program into your calculator:

- If a classmate has the program on her calculator
(any model TI-83/84), she
can transfer it to yours using the short cable with headphone-style
plugs that comes with all TI
calculators. On your calculator, press [
`2nd`

`x,T,θ,n`

*makes*`LINK`

] [`►`

] [`ENTER`

], and then on hers press [`2nd`

`x,T,θ,n`

*makes*`LINK`

] [`3`

], select VECPRODS, then press [`►`

] [`ENTER`

]. If you get a prompt about a duplicate program, choose Overwrite. - Or, download VECPRODS.ZIP (21 KB, updated 18 Sep 2008), unzip it, and transfer file VECPRODS.8XP to your calculator. (This requires TI-Connect or TI-Graph Link software and a cable.) If you get a prompt about a duplicate program, choose Replace.
- Or, as a last resort, key in the program. See VECPRODS.PDF and VECPRODS_HINTS.HTM in the VECPRODS.ZIP file.

However you get it into your calculator, run the program VECPRODS.

**Example** (with my thanks to Jason Duguay):
Find the dot product and cross product of

**u** = [0.894, 0.447, 0]
and
**v** = [−600, 200, −300]

As soon as you run the program, it
prompts you for the three components of each vector.
(**If your vectors have only x and y components**,
enter 0 for the z components.)
The program then displays two screens of information about the two
vectors and their products.

You’ll see the following information, in order:

- Vector
**u**in component form, confirming your input - The
**length or magnitude of**, symbolized u or ||**u****u**|| - Vector
**v**in component form - The
**length or magnitude of**, symbolized v or ||**v****v**|| - The
**dot product (inner product or scalar product)**of the two vectors,**u·v** - The
**angle θ between the vectors**, in degrees or radians according to the setting on your MODE screen

(This comes from the dot product:**u·v**= ||**u**|| ||**v**|| cos θ and therefore θ = cos^{-1}(**u·v**/(||**u**|| ||**v**||)).) - The
**cross product (outer product or vector product)**of the two vectors,**u**×**v**

(If your vectors**u**and**v**are in the xy plane, the cross product is parallel to the z axis. You can find more about the cross product below.) - The
**magnitude of the cross product**, which is the**area of a parallelogram**whose sides are vectors**u**and**v**(The area of the triangle with sides**u**and**v**is half the area of the parallelogram.)

The program stores its results in several variables, which are left afterward for your use:

- Lists LU and LV are your two input vectors.
- U and V are their magnitudes or lengths.
- X is
**u**·**v**. - List LW is
**u**×**v**.

To access any of them, press the [`ALPHA`

] key and then
the key for the letter such as [`ALPHA`

`6`

*makes* `V`

], or press
[`2nd`

`STAT`

*makes* `LIST`

] and then croll up to find the desired list name.

If you want to delete a list, press [`2nd`

`+`

*makes* `MEM`

]
[`2`

] [`4`

], cursor to each one, and press [`DEL`

].
To delete an ordinary variable, press [`2nd`

`+`

*makes* `MEM`

]
[`2`

] [`2`

], cursor to each one, and press [`DEL`

].

The program stores your graphics settings in
`GDB0`

and then automatically deletes that variable after
restoring your settings. You don’t care about this unless you're
using `GDB0`

for your own purposes, which would be quite
unusual.

Any two nonzero vectors **u** and **v** determine a
unique plane, assuming they’re not parallel.

The cross product **u×v** is a third vector, which is
defined in two ways as shown at right. Its magnitude is
||**u**|| ||**v**|| sin θ, the magnitude
of the first times the magnitude of the second times the sine of the
angle between them – this is also the area of a parallelogram
whose sides are **u** and **v**.
The cross product vector is normal (perpendicular) to the
plane containing the two vectors, indicated by the unit normal vector
**n**.

But *which* unit normal vector, since there are two?
(Think of one pointing above the plane and one pointing below.) The
answer is the infamous **right-hand rule**: if you hold your right
hand so that the fingers curl from the first vector toward the second,
then your thumb will point in the direction of the cross product vector
**u×v**.

How do you evaluate the cross product, in component form? The answer is the determinant you see above. Doesn’t help? There are two main ways to evaluate a 3-by-3 determinant:

The way I like best is to rewrite the first two columns to the right of the determinant and then take the six products shown:

The products going down to the right have a plus sign, and those going up to the right have a minus sign. This is Sarrus’s rule, due to the Frenchman J.P. Sarrus (1789–1861).

Some people prefer to evaluate the determinant by minors, the method of Pierre Simon, Marquis de Laplace (1749–1827). Remember that the second minor has its columns in reverse order from the original determinant!

Naturally, the two methods always give the same result (barring computational errors). Here’s a manual computation for the same cross product that the TI-83 or TI-84 calculated earlier:

=
[5(−1)−4(−7)]**i** + [(−7)((−6)−3(−1)]**j** + [3(4)−(−6)5]**k** =
23**i** + 45**j** + 42**k**

**31 Dec 2013**: Remove dead link to Irina Nelson and Johnny Erickson’s page showing the right-hand rule. (It’s still available here at the Internet Wayback Machine.)**18 Sep 2008**: Change program to display its output on two screens, with each vector component on a separate line to avoid truncation; remove separate program VECTORIN; use Jason Duguay’s example; add Program Variables section.**22 Jun 2008**: Remove source code because it’s now in the download file; standardize instructions for getting the program.- (intervening changes suppressed)
**24 Jul 2005**: New document.

This page is used in instruction at Tompkins Cortland Community College in Dryden, New York; it’s not an official statement of the College. Please visit www.tc3.edu/instruct/sbrown/ to report errors or ask to copy it.

For updates and new info, go to http://www.tc3.edu/instruct/sbrown/ti83/