How to Multiply Vectors on TI-83/84
Copyright © 2005–2023 by Stan Brown, BrownMath.com
Copyright © 2005–2023 by Stan Brown, BrownMath.com
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 −7a. 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 = u1v1 + u2v2; in space it’s u1v1 + u2v2 + u3v3.
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 ] |
The VECPRODS program (below) will also compute the dot product of two vectors.
You know that the length or magnitude of vector v is found by
||v|| = √v1² + v2²
You know also that the dot product of a vector with itself is
v·v = v1v1 + v2v2 = v1² + v2²
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 ] |
The VECPRODS program (below) will also compute the lengths of two vectors.
I wrote a 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.
The program works on all TI-83s and TI-84s, including the newer color models.
There are three methods to get the program into your calculator:
2nd
x,T,θ,n
makes LINK
]
[►
] [ENTER
]. Then on hers press
[2nd
x,T,θ,n
makes LINK
] [3
], select
VECPRODS,
and finally press [►
] [ENTER
].
If you get a prompt about a duplicate program, choose Overwrite.
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:
The program stores its results in several variables, which are left afterward for your use:
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 scroll 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 = 23i + 45j + 42k
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