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Converting between Cartesian and Vector form
Fichas del conjunto (11)
- 1An example of a 2D Cartesian equation of r=a+λb is....y = mx + c
(where a=[a1][a2] and b=[b1][b2]) - 2An example of a 3D Cartesian equation of r=a+λb is....(x-a1)/b1 = (y-a2)/b2 = (z-a3)/b3
- 3In 2D, a line with the gradient p/q has the direction vector....[q][p]
- 4E.g. A line passes through (3,-5) and has the gradient -2/7 . Find the vector equation of this lineDirection vector = [7][-2] (draw diagram to help)
Point on line = [3][-5]
r = [3][-5] + λ [7][-2] - 5Write r = [1][2] + λ [5][3] in form ax+by+c=0[x][y] = [1][2] + λ [5][3]
Equate x's: x=1+5λ , λ=(x-1)/5
Equate y's: y=2+3λ , λ=(y-2)/3
Set equal : (x-1)/5 = (y-2)/3
Rearrange to required form: 3x-5y+7=0 - 6Write r = [1][4][-1] + λ [3][2][5] in Cartesian form[x][y][z] = [1][4][-1] + λ [3][2][5]
Equate x's: x=1+3λ , λ=(x-1)/3
Equate y's: y=1/2+5λ , λ=(y-4)/2
Equate z's: z=-1+5λ , λ=(z+1)/5
Equation : (x-1)/3 = (y-4)/2 = (z+1)/5
(Could also use (x-a1)/b1 = (y-a2)/b2 = (z-a3)/b3 if not "show that") - 7Write r =[1][1/2][-3] + λ [1/3][5][0] in Cartesian formUsing (x-a1)/b1 = (y-a2)/b2 = (z-a3)/b3
(x-1)/1/3 = (y-1/2)/5 , z=-3
(z=-3 as we cant divide -3 by 0)
Same as: 3x-3 = (2y-1)/10 , z=-3 - 8A comma followed by "z=-3" means what?Z never moves from -3
- 9Find the vector equation of : x=-2 , (3y+1)/4 = (2-z)/5As x=-2 , we immediately know :
r=[-2][-][-] + λ [0][-][-]
Next we say: λ = (3y+1)/4 and λ = (2-z)/5
These rearrange to: y=4/3λ-1/3 and z=2-5λ
Therefore: r = [-2][-1/3][2] + λ [0][4/3][-5] - 10Determine whether A(3,-2,2) lies on : (x+1)/2 = (4-y)/3 = 2z/3Sub in: 3+1/2 = 4--2/3 = 2(2)/3
Same as 2=2≠4/3
Therefore doesn't lie on the line - 11Write 3x-5y=17 in vector form3x-5y=17 is same as y=3/5x - 17/5
3/5x is same as direction vector [5][3]
To find point, set x=0 so therefore y= intercept (-17/5)
r = [0][-17/5] + λ [5][3]
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- Converting between Cartesian and Vector form
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- Doc Boff
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