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

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 Q͞P⃗, 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 −2i + 2j (where i = ⟨1, 0⟩ and j = ⟨0, 1⟩ in 2 dimensions), so you can also write (2, 3) as O + 2i + 3j (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

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:

⟨a, b⟩ × ⟨c, d⟩ = ad − bc.

For example, ⟨−2, 2⟩ × ⟨3, 1⟩ = (−2)(1) − (2)(3) = −8. Geometrically,

u × v = |u| |v| sin∠̄(u, v),

where ∠̄(u, v), the signed measure of the angle from u to v, is positive if this angle is counterclockwise and negative if it's clockwise. (At least that's so using a counterclockwise coordinate system, which is like using a right-handed coordinate system in 3 dimensions.)

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

×⟨c, d⟩ = ⟨d, −c⟩.

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

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