This post covers features of a point in R2 and two methods for the multiplication of points. The first method is to multiply the R2 coordinates directly. The second method is to use multiplication formulae based on R2 trigonometry.
Features of a Point
The distance R of any point (a, b) from the origin is given by Pythagoras’ theorem and is R = √(|a|2 + |b|2).
For example, the distance R of the point (3^,2i^) from the origin is √(|3|2 + |2|2) = √14 = 3.606. This is a Counter because distance is a magnitude.
A line from the point (3^,2i^) that forms a right angle with the y-axis has a length of 3. The point (3^,2i^) is on a circle of radius 3.606 with its centre at the origin. Furthermore, the line from (3^,2i^) to the y-axis forms a right-angle triangle with the radius and the y-axis. The length of the triangle side on the y-axis is 2.
Similarly, a line from the point (3^,2i^) that forms a right angle with the x-axis has a length of 2. The point (3^,2i^) is on a circle of radius 3.606 with its centre at the origin. Furthermore, the line from (3^,2i^) to the x-axis forms a right-angle triangle with the radius and the x-axis. The length of the triangle side on the x-axis is 3.
φx is the smaller angle between the x-axis and the point and its cosine is 3/3.606 = 0.832.
φy is the smaller angle between the point and the y-axis and its cosine is 2/3.606 = 0.555.
Multiplication of Two Points
Take that the points A and B at (Ax, Ay) and (Bx, By) are to be multiplied to give a result C at (Cx, Cy). The first method of doing this is just to multiply the Cartesian coordinates together as:
(Ax + Ay)*(Bx + By) = Ax*Bx + Ax*By + Ay*Bx* + Ay*By
Alternatively, use an R2 formula with trigonometric functions that multiplies both the magnitude of a point and the angles of the the points. This is done using the following formula:
- Cx = RA*RB(cos(φAx)*1‡Ax*cos(φBx)*1‡Bx + cos(φAy)*1‡Ay*cos(φBy)*1‡By)
- Cy = RA*RB(cos( φAx)*1‡Ax*cos(φBy)*1‡By + cos(φAy)*1‡Ay*cos(φBx)*1‡Bx)
Here RA is the distance of (Ax, Ay) from the origin and RB is the distance of (Bx, By) from the origin. φAx, φAy, φBx and φBy correspond to the angles in the diagram above for each of the points A and B,
Example 1
Let A = (3^, 2i^) and B = (^, 4i^):
Using the Multiplication of 2 points method
Multiplication of these 2 points gives the following:
- A*B = 3^*^ +3^*4i^ + 2i^*^+ 2i^*4i^
- = 3^+ 12i^ + 2i^+ 8v
- = 5v+ 14i^
Try this example of multiplication with our online calculator.
Using the R2 Trigonometric Formula for Multiplication
Alternatively, the R2 formula from trigonometry can be used for multiplication:
First, calculate the radii:
- RA = √((|3|2 + |2|2)) = √13 = 3.606
- RB = √((|1|2 + |4|2)) = √17 = 4.12
Next, calculate the cosines:
- cos(φAx) = 3/3.606 = 0.832
- cos(φAy) = 2/3.606 = 0.555
- cos(φBx) = 1/4.12 = 0.243
- cos(φBy) = 4/4.12 = 0.97
Finally, apply the formula:
- Cx = 3.606*4.12( 0.832^*0.243^ + 0.555i^*0.97i^)
- = 14.857*(0.202^ + 0.539v) = 5v
- Cy = 3.606*4.12( 0.832^*0.97i^ + 0.555i^*0.243^)
- = 14.857*(0.808i^ + 0.135i^) = 14i^
Example 2
Let A = (3v, 2i^) and B = (^, 4iv):
Using the Multiplication of 2 points method
Multiplication of these 2 points gives the following:
- A*B = 3v*^ +3v*4iv + 2i^*^+ 2i^*4iv
- = 3v+ 12i^ + 2i^+ 8^
- = 5^+ 14i^
Using the R2 Trigonometric Formula for Multiplication
Alternatively, the R2 formula from trigonometry can be used for multiplication using the same radii and cosines as in the previous example because the Counters are identical for each of the points:
- Cx = 3.606*4.12( 0.832v*0.243^ + 0.555i^*0.97iv)
- = 14.857*(0.202v + 0.539^) = 5^
- Cy = 3.606*4.12( 0.832v*0.97iv + 0.555i^*0.243^)
- = 14.857*(0.808i^ + 0.135i^) = 14i^
Example 3
Let A = (3v, 2iv) and B = (v, 4iv):
Using the Multiplication of 2 points method
Multiplication of these 2 points gives the following:
- A*B = 3v*v +3v*4iv + 2iv*v+ 2iv*4iv
- = 3^+ 12i^ + 2i^+ 8v
- = 5v+ 14i^
Example 4
Let A = (3v, 2i^) and B = (v, 4iv):
Using the Multiplication of 2 points method
Multiplication of these 2 points gives the following:
- A*B = 3v*v +3v*4iv + 2iv*v+ 2iv*4iv
- = 3^+ 12i^ + 2iv+ 8^
- = 11^+ 10i^
Using the R2 Trigonometric Formula for Multiplication
Alternatively, the R2 formula from trigonometry can be used for multiplication using the same radii and cosines as in the previous example because the Counters are identical for each of the points:
- Cx = 3.606*4.12( 0.832v*0.243v + 0.555i^*0.97iv)
- = 14.857*(0.202^ + 0.539^) = 11^
- Cy = 3.606*4.12( 0.832v*0.97iv + 0.555i^*0.243v)
- = 14.857*(0.808i^ + 0.135iv) = 10i^
An Alternative R2 Trigonometric Formula for Multiplication
An alternative formula based on classical math is to add the two angles φx and φx to get the new location of the point. The angle φx in this formula represents the full angle between the x-axis and the point. The angle φy represents the full angle between the point and the y-axis. The formula is adapted from the one for complex numbers multiplication as outlined in Wikipedia.
- Cx = RA*RB(cos(φAx + φBx))^
- Cy = RA*RB(sin(φAx + φBx))i^
The examples below are the same as those above. In all these examples:
- cos(φAx) = 0.832 => φAx = 33.7^o
- cos(φBx) = 0.243 => φBx= 75.95^o
- => φAx + φBx = 109.65^o
Using the alternative formula, here are the calculations:
Ex. 1 – A = (3^, 2i^) and B = (^, 4i^):
Given φAx + φBx = 109.65^o:
- Cx = RA*RB(cos(φAx + φBx))^ = 14.857(cos(109.65^o)^ = 14.857(–0.336)^ = 5v
- Cy = RA*RB(sin(φAy + (φBy))i^ = 14.857(sin(109.65^o)i^ = 14.857(0.942)^ = 14i^
Ex. 2 – A = (3v, 2i^) and B = (^, 4iv):
The point A is in the second quadrant and now the counterclockwise angle from the x-axis is 180^o less 33.7^o giving φAx equal to 146.3^o. The point B is in the fourth quadrant and now the counterclockwise angle from the x-axis is 360^o less 75.95^o giving φBx equal to 284.05^o. The sum of the two angles is 146.3^o + 284.05^o = 430.35^o or 70.35^o. So:
- Cx = RA*RB(cos(φAx + φBx))^ = 14.857(cos(70.35^o)^ = 14.857(0.336)^ = 5^
- Cy = RA*RB(sin(φAy + (φBy))i^ = 14.857(sin(70.35^o)i^ = 14.857(0.942)^ = 14i^
Ex. 3 – A = (3v, 2iv) and B = (v, 4iv):
The point A is in the third quadrant and now the counterclockwise angle from the x-axis is 180^o plus 33.7^o giving φAx equal to 213.7^o. The point B is also in the third quadrant and now the counterclockwise angle from the x-axis is 180^o plus 75.95^o giving φBx equal to 255.95^o. The sum of the two angles is 213.7^o + 255.95^o = 469.65^o or 109.65^o. So:
- Cx = RA*RB(cos(φAx + φBx))^ = 14.857(cos(109.65^o)^ = 14.857(–0.336)^ = 5v
- Cy = RA*RB(sin(φAy + (φBy))i^ = 14.857(sin(109.65^o)i^ = 14.857(0.942)^ = 14i^
Ex. 4 – A = (3v, 2i^) and B = (v, 4iv):
The point A is in the second quadrant and now the counterclockwise angle from the x-axis is 180^o less 33.7^o giving φAx equal to 146.3^o. The point B is in the third quadrant and now the counterclockwise angle from the x-axis is 180^o plus 75.95^o giving φBx equal to 255.95^o. The sum of the two angles is 146.3^o + 255.95^o = 402.25^o or 42.25^o. So:
- Cx = RA*RB(cos(φAx + φBx))^ = 14.857(cos(42.25^o)^ = 14.857(0.740)^ = 11^
- Cy = RA*RB(sin(φAy + (φBy))i^ = 14.857(sin(42.25^o)i^ = 14.857(0.672)^ = 10i^
Conclusion
The results from the formulae match the results of the direct multiplication.
Next: Dot Product
Previous: Simple Multiplication