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:
 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 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⟩. 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).

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

NameSymbolResult Order matters?Depends on lengths? Depends on orientation?
Dot product ScalarNo YesNo
Cross product× Scalar in 2D,
vector in 3D
YesYes 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 counterclockwise orientation, which is like using the right-handed orientation in 3 dimensions):

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 measure of the angle from u to v, is positive if this angle is counterclockwise (using the counterclockwise orientation again) and negative if it's clockwise.

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

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