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〉 (which you can also write as −2i + 2j).
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?|
|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 (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. For example, 〈−2, 2〉 × 〈3, 1〉 = (−2)(1) − (2)(3) = −8.
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. 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.
This web page was written in 2014 and 2016 by Toby Bartels, last edited on 2016 January 7. Toby reserves no legal rights to it.
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