Add: | Scalar, | Point, | Vector. | Subtract: | Scalar, | Point, | Vector. | Multiply: | Scalar, | Point, | Vector. | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Scalar: | Scalar | ― | ― | Scalar | ― | ― | Scalar | ― | Vector | |||

Point: | ― | ― | Point | ― | Vector | Point | ― | ― | ― | |||

Vector: | ― | Point | Vector | ― | ― | Vector | Vector | ― | Depends |

The operations involving points aren't in the textbook,
but they work just like the operations involving vectors.
If *P* is a point and **v** is a vector,
then *P* + **v** is the point obtained
by starting at *P*
and moving in the direction and distance given by **v**.
For example, (2, 3) + ⟨2, 1⟩ =
(2 + 2, 3 + 1) = (4, 4);
the general formula (in 2 dimensions) is
(*a*, *b*) +
⟨*c*, *d*⟩ =
(*a* + *c*, *b* + *d*).
Similarly, if *P* and *Q* are points,
then *P* − *Q* is the vector,
sometimes denoted *Q͞**P⃗*,
giving the direction and distance to start at *Q* and arrive at *P*.
For example, (2, 3) − (4, 1) =
⟨2 − 4, 3 − 1⟩ =
⟨−2, 2⟩;
the general formula (in 2 dimensions) is
(*a*, *b*) −
(*c*, *d*) =
⟨*a* − *c*, *b* − *d*⟩
for the vector from (*c*, *d*) to (*a* *b*).

Just as you can write ⟨−2, 2⟩
as −2**i** + 2**j**
(where **i** = ⟨1, 0⟩
and **j** = ⟨0, 1⟩
in 2 dimensions),
so you can also write (2, 3)
as *O* + 2**i** + 3**j**
(where *O* = (0, 0) in 2 dimensions).
All of the usual algebraic rules apply
for addition, subtraction, and scalar multipltication.

There are various ways to multiply two vectors, with results as in this table:

Name | Symbol | Result | Order matters? | Depends on lengths? | Depends on orientation? |
---|---|---|---|---|---|

Dot product | ⋅ | Scalar | No | Yes | No |

Cross product | × | Scalar in 2D, vector in 3D |
Yes | Yes | Yes |

The cross product in 2 dimensions is not in the textbook; here is the formula for it:

⟨For example, ⟨−2, 2⟩ × ⟨3, 1⟩ = (−2)(1) − (2)(3) = −8. Geometrically,a,b⟩ × ⟨c,d⟩ =ad−bc.

where ∠̄(u×v= |u| |v| sin∠̄(u,v),

Recall that subtraction is adding the opposite:
*a* − *b* = *a* + (−*b*),
and **u** − **v** = **u** + (−**v**).
Similarly, the cross product in 2 dimensions (but *not* in 3 dimensions)
can be done using the dot product and a *rotation*:
**u** × **v** = **u** ⋅ (×**v**),
where ×**v** is obtained from **v**
by rotating it clockwise through a right angle
(assuming a counterclockwise coordinate system again).
The formula for this
is

×⟨For example, ×⟨3, 1⟩ = ⟨1, −3⟩, so ⟨−2, 2⟩ × ⟨3, 1⟩ = ⟨−2, 2⟩ ⋅ ⟨1, −3⟩ = (−2)(1) + (2)(−3) = −8 again.c,d⟩ = ⟨d, −c⟩.

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This web page was written between 2014 and 2019 by Toby Bartels, last edited on 2019 April 1. Toby reserves no legal rights to it.

The permanent URI of this web page
is
`http://tobybartels.name/MATH-2080/2020SP/vecops/`

.