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); 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).
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?|
|Cross product||×||Scalar in 2D,
vector in 3D
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 measure of the angle from u to v, is positive if this angle is counterclockwise (using a counterclockwise coordinate system, which is like using a right-handed coordinate system in 3 dimensions) 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 (using 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.
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