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 book,
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).
Similarly, if *P* and *Q* are points,
then *P* − *Q* is the vector,
sometimes denoted *Q*͞*P*⃗,
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⟩.
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).

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 (using the counterclockwise orientation, which is like using the right-handed orientation in 3 dimensions):

⟨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
(again using the counterclockwise orientation).
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.c,d⟩ = ⟨d, −c⟩.

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This web page was written in 2014 and 2016 by Toby Bartels, last edited on 2016 March 30. Toby reserves no legal rights to it.

The permanent URI of this web page
is
`http://tobybartels.name/MATH-2080/2016SP/vecops/`

.