Operations involving vectors

Vector algebra involves several different types of values; in this course, we study scalars, points, and vectors. We can try to add these, subtract them, and multiply them in various ways. Sometimes these operations make sense, and sometimes they don't, with results as in this table:

	Scalar,	Point,	Vector.	Scalar,	Point,	Vector.	Scalar,	Point,	Vector.
Scalar:	Scalar	―	―	Scalar	―	―	Scalar	―	Vector
Point:	―	―	Point	―	Vector	Point	―	―	―
Vector:	―	Point	Vector	―	―	Vector	Vector	―	Depends

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

Name	Symbol	Result	Order matters?	Depends on lengths?	Depends on the right-hand rule?
Dot product	⋅	Scalar	No	Yes	No
Cross product	×	Scalar in 2D, vector in 3D	Yes	Yes	Yes

There are actually other ways to multiply vectors besides these, but we don't cover them in this course.

The cross product in 2 dimensions is not in the textbook; here is the formula for it (using the right-hand rule):

⟨a, b⟩ × ⟨c, d⟩ = ad − bc.

Geometrically,

u × v = |u| |v| sin∠(u, v),

where ∠(u, v), the measure of the angle from u to v, is positive if this angle is counterclockwise (using the right-hand rule) and negative if it's clockwise.

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This web page was written in 2014 and 2015 by Toby Bartels, last edited on 2015 April 26. Toby reserves no legal rights to it.

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