Let A = (2,7), B = (-1,4) and C = (3,0) be points on the plane...
Let A = (2,7), B = (-1,4) and C = (3,0) be points on the plane
Let l be the line whose equation is x+ky=0 for some real number k.
Find all values of k so that l meet the triangle ABC.
Homework Answers
28 Consider (O:OAOB) an orthonormal system in space. Let G be the center of gravity of triangle ABC. 1° Calculate the coordinates of G 2°Consider the points A' (2 ;0:0) ,B, (0:2:0) and C"...
28 Consider (O:OAOB) an orthonormal system in space. Let G be the center of gravity of triangle ABC. 1° Calculate the coordinates of G 2°Consider the points A' (2 ;0:0) ,B, (0:2:0) and C" (0:0,3). a) Verify that these three points define a plane. b) Write a system of parametric equations of the plane (A'BC'). 3 Write a system of parametric equations of line (AC). 4° Verify that K (4:0-3) is the trace of the line (AC) with the plane...
The Moulton Plane is the plane M = (R2, LM) such that a subset I of...
The Moulton Plane is the plane M = (R2, LM) such that a subset I of R2 belongs to LM if and only if one of the following holds: i) l = {(x,y)| x=a} (vertical line); ii) l = {(x,y)| y=b} (horizontal line) iii) ( = {(x,y)| y = mx +b where m<0} (line with negative slope) [ m(x - x0) if x xo when m>0}. (bent line W 14,9 m ( x - x0) if x > xo with...
Let C be a triangle in the x-y plane with vertices (x1,y1), (x2y2) and (x3,y3) arranged so that C...
Let C be a triangle in the x-y
plane with vertices (x1,y1), (x2y2) and (x3,y3) arranged so that C
is positively-oriented.
Let C be a triangle in the xy-plane with vertices (x,y), (z2,p), and (z3,U3) arranged so that C is positively-oriented. a.) Sketch such a triangle and indicate its orientation. b.) Apply Green's Theorem to compute the area of the triangle as a (sum of) path integral(s) around the boundary. Get a formula for area in terms of the coordinates...
The plane passing through the 3 points A(4, -2, 1), B(4, -1,4) and C(1,-4,2) can be...
The plane passing through the 3 points A(4, -2, 1), B(4, -1,4) and C(1,-4,2) can be expressed as a parametric vector equation of the form x a 1V+ 12v2 where 1 and d2 are real parameters. You are given that V1 Find suitable vectors a and V2 and enter them in the boxes below. You must enter your answers using Maple notation, for example, the vector \(\begin{pmatrix}1\2\\3\end{pmatrix} is entered as <1, 2, 3>. a = Your response Correct response No...
Q1. Given the points A: (0,0,2), B: (3,0,2), C: (1,2,1), and D: (2, 1,4 a) Find the cross product v - AB x AC. b) Find the equation of the plane P containing the triangle with vertices A, B, and C c)...
Q1. Given the points A: (0,0,2), B: (3,0,2), C: (1,2,1), and D: (2, 1,4 a) Find the cross product v - AB x AC. b) Find the equation of the plane P containing the triangle with vertices A, B, and C c) Find u the unit normal vector to P with direction v d) Find the component of AD over u and the angle between AD and u, then calculate the volume of the parallelepiped with edges AB, AC, AD...
1. (10) Let l be the line in 3-space that passes through the points A=(5,2, -1)...
1. (10) Let l be the line in 3-space that passes through the points A=(5,2, -1) and B = (6,0,–7). (a) Find a set of parametric equations for l. (b) Find the unique point P at which l intersects the plane with equation -3.21 + 722 - 2.23 = 11. (c) Let P be the point found in part (b), and let Q = (k, 7, 10) for an unspecified real number k. Determine the value of k for which...
please provide explanations. (a) (7 points) Use the Green's Theorem to evaluate the line integral y dr+ry dy, wh...
please provide explanations.
(a) (7 points) Use the Green's Theorem to evaluate the line integral y dr+ry dy, where 2 C is the positively oriented triangle with vertices (0,0), (2,0) and (2,6) (b) (7 points) Let F(x, y) = (2xsin(y) + y2) i(x2 cos(y) +2ry)j. Find the scalar function f such that Vf F. equation of the tangent plane to the surface r(u, v) (u+v)i+3u2j+ (c) (7 points) Find an (u- v) k at the point (ro, yo, 20) (2,...
Let a, b, c, d and e be the first five digits of your La Trobe...
Let a, b, c, d and e be the first five digits of your La Trobe student number. i.e. if your student number is 12345678 then a = 1,b=2,c=3, d = 4 and e = 5. Use your appropriate values in the question below. 2 a Consider the line. L = {(2,3, 2): ?- } y-b 3 2 (a) Write L in parametric form. (b)0 Find the coordinates of the point P at which L intersects the plane z =...
5. (a) Let u 1,4,2), ,1,0). Find the orthogonal projection of u on v (b) Letu...
5. (a) Let u 1,4,2), ,1,0). Find the orthogonal projection of u on v (b) Letu ,1,0), u(0,1,1), (10,1). Find scalars c,,s such that 6. (a) Find the area of the triangle with vertices , (2,0,1), (3, 1,2). Find a vector orthogonal to the plane of the triangle. (b)) Find the distance between the point (1,5) and the line 2r -5y1 (i) Find the equation of the plane containing the points (1,2, 1), (2,1, 1), (1, 1,2). 7. (a) Let...
Given in space the points A(4,7,1), B(2,1,3), and c(0,-1,2) The vectors ū = AB , and...
Given in space the points A(4,7,1), B(2,1,3), and c(0,-1,2) The vectors ū = AB , and ✓ = AC a. (9%) Find ū. v , ū x ū , proj, u b. (3%) Find the area of triangle ABC. c. (3 %) Find the parametric equation of line (AB). d. (3 %) Find the distance from point C to the line (AB). e. (3 %) Find the equation of the plane (ABC). A relatively easy way of getting into international...
28 Consider (O:OAOB) an orthonormal system in space. Let G be the center of gravity of triangle ABC. 1° Calculate the coordinates of G 2°Consider the points A' (2 ;0:0) ,B, (0:2:0) and C" (0:0,3). a) Verify that these three points define a plane. b) Write a system of parametric equations of the plane (A'BC'). 3 Write a system of parametric equations of line (AC). 4° Verify that K (4:0-3) is the trace of the line (AC) with the plane...
The Moulton Plane is the plane M = (R2, LM) such that a subset I of R2 belongs to LM if and only if one of the following holds: i) l = {(x,y)| x=a} (vertical line); ii) l = {(x,y)| y=b} (horizontal line) iii) ( = {(x,y)| y = mx +b where m<0} (line with negative slope) [ m(x - x0) if x xo when m>0}. (bent line W 14,9 m ( x - x0) if x > xo with...
Let C be a triangle in the x-y
plane with vertices (x1,y1), (x2y2) and (x3,y3) arranged so that C
is positively-oriented.
Let C be a triangle in the xy-plane with vertices (x,y), (z2,p), and (z3,U3) arranged so that C is positively-oriented. a.) Sketch such a triangle and indicate its orientation. b.) Apply Green's Theorem to compute the area of the triangle as a (sum of) path integral(s) around the boundary. Get a formula for area in terms of the coordinates...
The plane passing through the 3 points A(4, -2, 1), B(4, -1,4) and C(1,-4,2) can be expressed as a parametric vector equation of the form x a 1V+ 12v2 where 1 and d2 are real parameters. You are given that V1 Find suitable vectors a and V2 and enter them in the boxes below. You must enter your answers using Maple notation, for example, the vector \(\begin{pmatrix}1\2\\3\end{pmatrix} is entered as <1, 2, 3>. a = Your response Correct response No...
Q1. Given the points A: (0,0,2), B: (3,0,2), C: (1,2,1), and D: (2, 1,4 a) Find the cross product v - AB x AC. b) Find the equation of the plane P containing the triangle with vertices A, B, and C c) Find u the unit normal vector to P with direction v d) Find the component of AD over u and the angle between AD and u, then calculate the volume of the parallelepiped with edges AB, AC, AD...
1. (10) Let l be the line in 3-space that passes through the points A=(5,2, -1) and B = (6,0,–7). (a) Find a set of parametric equations for l. (b) Find the unique point P at which l intersects the plane with equation -3.21 + 722 - 2.23 = 11. (c) Let P be the point found in part (b), and let Q = (k, 7, 10) for an unspecified real number k. Determine the value of k for which...
please provide explanations.
(a) (7 points) Use the Green's Theorem to evaluate the line integral y dr+ry dy, where 2 C is the positively oriented triangle with vertices (0,0), (2,0) and (2,6) (b) (7 points) Let F(x, y) = (2xsin(y) + y2) i(x2 cos(y) +2ry)j. Find the scalar function f such that Vf F. equation of the tangent plane to the surface r(u, v) (u+v)i+3u2j+ (c) (7 points) Find an (u- v) k at the point (ro, yo, 20) (2,...
Let a, b, c, d and e be the first five digits of your La Trobe student number. i.e. if your student number is 12345678 then a = 1,b=2,c=3, d = 4 and e = 5. Use your appropriate values in the question below. 2 a Consider the line. L = {(2,3, 2): ?- } y-b 3 2 (a) Write L in parametric form. (b)0 Find the coordinates of the point P at which L intersects the plane z =...
5. (a) Let u 1,4,2), ,1,0). Find the orthogonal projection of u on v (b) Letu ,1,0), u(0,1,1), (10,1). Find scalars c,,s such that 6. (a) Find the area of the triangle with vertices , (2,0,1), (3, 1,2). Find a vector orthogonal to the plane of the triangle. (b)) Find the distance between the point (1,5) and the line 2r -5y1 (i) Find the equation of the plane containing the points (1,2, 1), (2,1, 1), (1, 1,2). 7. (a) Let...
Given in space the points A(4,7,1), B(2,1,3), and c(0,-1,2) The vectors ū = AB , and ✓ = AC a. (9%) Find ū. v , ū x ū , proj, u b. (3%) Find the area of triangle ABC. c. (3 %) Find the parametric equation of line (AB). d. (3 %) Find the distance from point C to the line (AB). e. (3 %) Find the equation of the plane (ABC). A relatively easy way of getting into international...
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