Prove that the centroid of a parallelogram is the point of intersection of the diagonals of the parallelogram. [Hint: Choose coordinates so that the vertices of the parallelogram are located at (0, 0), (0, a), (b, c), and (b, a + c).]

Q: (a) Find and identify the traces of the quadric surface x2 + y2 - z2 = 36 given the plane.

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Q: Show that the points A = (1, 2), B = (1,0), and C = collinear and express P = (-7,0) in barycentric…

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Q: Find the coordinates of the centroid of the triangle with the given vertices. X(1, 4), Y(7, 2), Z(2,…

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Q: Determine the Miller indices of a plane that makes intercepts of (4, 3 and 8) on the coordinate axes…

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Q: Find the centroid (ã, §) of the triangle with vertices at (0, 0), (2, 0), and (0, 8). || 18

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Q: Find the centroid (ĩ, g) of the triangle with vertices at (0, 0), (10, 0), and (0, 9).

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Q: C, D are coplanar points. A', B', C', D' are their projections respectively on the y-

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Q: Draw an obtuse triangle and, by construction, find the centroid of the triangle. (HINT: Begin by…

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Q: prove that the system of parabolas y2=4a(x+a) is self orthogonal

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Q: Find the angles between the planes in Exercises 47 and 48. 47. x + y = 1, 2x + y - 2z = 2 48. 5x + y…

A: Hey, since there are multiple questions posted, we will answer the first question. If you want any…

Q: For a plane with a normal of n = [1, 2, - 9]   and containing a point P (3, -4, 0), write the…

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Q: Every triangle has a balancing point, called the centroid. O True O False

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Q: Find the distance from y to the plane in R spanned by the set {u1, uz} 4 - 3 -3 y = -9 -4 Uz 2 4 1…

A: Solution   Given,       Find the distance from  γ to the plane  in R3       spanned  by the  set…

Q: The figure shows a parallelepiped spanned by vectors a,b and c Which of the following formulas…

A: The formula to calculate the volume is given by, Volume=area of base⋅height             =∥b×c∥ ∥a∥…

Q: Prove that the planes determined by the equations ax + by + c z = d and αx + β y + γ z = δ are…

A: Given planes, ax + by + c z = d αx + β y + γ z = δ

Q: 1. Find the area of the triangle in the R2 plane whose vertices are given by (0,0), (–1,4) and (7,…

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Q: Show that the system of confocal and co-axial parabolas y²-4a(a+x) is self orthogonal, where a is…

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Q: Use the triple scalar product to find the volume of the parallelepiped having adjacent edges u, v,…

A: Find the volume of parallelepiped

Q: prove that the system of parabolas y² = 4a(x + a) is self orthogonal

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Q: Find the barycentric coordinates of p with respect to the affinely independent set of points that…

A: Given 11, 35, 06, p=23 let a=11, b=35, c=06, p=23 Now we write the points in augmented matrix form…

Q: Show there is no plane perpendicular to n=i+jn=i+j that passes through points P(1,2,3)P(1,2,3) and…

A: This is the rough image Point P(1,2,3) and Q(2,3,4) are the points on the plane…

Q: Find the symmetric equations of the L that is parallel to the two planes x+y-z= 2 and z + 3y+ z =…

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Q: If (-2, 3), (4, -3) and (4, 5) are the mid-points of the sides of a triangle, find the coordinates…

A: On solving this we will get:-

Q: In Exercises 17-20, find the equation of the plane with the given description.17. Passes through O…

A: Given, A plane passes through O and is parallel to 4x-9y+z=3

Q: is ndicular tò the plane containing the points: A = (-1,1, 2) B = (-3, 2, 1) C = (5 – 1, 1) the

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Q: Find an equation of the plane through the origin that is parallel to the plane 5x – 4y + 6z + 10 =…

A: Here we have to find the equation of a plane.

Q: In space coordinates, the xy plane consists of all the points of the form O (0, y, z) O (X, y, z) O…

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Q: Find the projection of a on the plane P if a 4 and P has the equation x + y + z = 1.

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Q: Prove that the shape ABCD is a rhombus. With A(1, 4), B(3, 0), C(1. -4) and D(-1,0).

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Q: Use the triple scalar product to find the volume of the parallelepiped having adjacent edges u, v,…

A: Given, u=<1,3,1> v=<0,6,6> w=<-4,0,-4>

Q: Find %3D as if F(x, y, z) = 2xi+2yj+2zk and S is the part of the paraboloid z = 4-x-y lying above…

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Q: find the centroids of the triangleswhose vertices appear . Assume a, b > 0. (0, 0), (1, 0), (0,…

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Q: Find the centroid of the triangle formed by joining the points A (3, 2), B (-1, 4), and C (5, -3).…

A: Explained below

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Q: Prove that the planes (i) 12x – 15y + 16z – 28 = 0, (ii) 6x + 6y – 7z –8 = 0, and (üi) 2x + 35y -…

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Q: Determine whether the planes are parallel, perpendicular, or neither. 2x – 10y – 8z – 1 = 0 (a) | -x…

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Q: Find the centroid of the parabolic spandrel shown in the figure.

A: We have given a diagram of parabolic spandrel;

Q: The centroid of a triangle lies at the intersection of the triangle's medians, because it lies…

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Q: Let D* be a parallelogram in the uv-plane with vertices A*(o)0), B*(를/듣), C*(를 ') E*( ).Let D be a…

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Q: Using the fact that the centroid of a triangle lies at the intersection of the triangle's medians,…

A: We can find the centroid as below using the required formula.

Q: Verify that the two planes are parallel, and find the distance between the planes. (Round your…

A: Using distance formula between parallel planes

Q: Find points in the Euclidean plane corresponding to the points e₁-[3,-5,-1], e2=[1,-2,2], e3 [2,1,0]…

A: By Pn , we denote the projective space of dimension n, defined as follows: Pn=Rn+1 \ {0}~, where '~'…

Q: Find the centroid (, j) of the triangle with vertices at (0, 0), (8, 0), and (0, 9). x=

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Q: Find the centroid (ã, g) of the triangle with vertices at (0, 0), (5, 0), and (0, 9). y=

A: In the question it is asked to find centroid of the triangle with coordinates (0,0), (5,0) and…

Q: Find a Find a funds on the surdace 7=372y8 at which the feugend Planes is Parallel to the Plane…

A: We have to find the value of points on the surface