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); 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? |
---|---|---|---|---|---|
Dot product | ⋅ | Scalar | No | Yes | No |
Cross product | × | Scalar in 2D, vector in 3D |
Yes | Yes | Yes |
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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