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 by Toby Bartels, last edited on 2020 August 24. Toby reserves no legal rights to it.

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