The equation of a plane n ·〈x, y,z〉= d is said to be in normal form if n is a unit vector. Show that in this case, |d| is the distance from the plane to the origin. Write the equation of the plane 4x − 2y + 4z = 24 in normal form.

Q: Find the equation of a plane that is perpendicular to the vector 4 i + 2 j - 2 k and passing through…

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Q: Find equations for the planes The plane through A(1, -2, 1) perpendicular to the vector from the…

A: The normal vector of the plane is given by A(1,-2,1)  The point of passing is (0,0,0). The equation…

Q: Find the vector, parametric, and symmetric equations of the line which passes through the points A =…

A: Given points are: A=3,1,-1 and B=7,0,1 To find the vector, parametric, and symmetric equations

Q: Use the component form to generate an equation for the plane through P1(4, 1, 5) normal to n1 = i -…

A: Recall the following fact.

Q: a) Find equation of the Line through point (2,3,0) perpendicular to the vectors U1 = i+2j+ 3k and U…

A: Line through point (2, 3,0)  Which perpendicular to U1 and U2

Q: Find a vector equation for the plane that contains the points P, Q and R Find a scalar equation for…

A: Given points: P(2,1,1),Q(0,4,1),R(−2,1,4) and S(1,5,−4) Equation of a plane in cartesian form…

Q: Find the distance of point A(2/3, 2) from the line that passes through the origin and parallel to…

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Q: Find the equation of a plane that is perpendicular to the vector – i+ 2j+ 2k and passing through the…

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Q: Is the line x = 6- 3t, y = 8 + 8t, z= -t parallel to the plane 3x + y -z=2? Give reasons for your…

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Q: write the equation of the plane passingthrough P with direction vectors u and v in (a) vector form…

A: A plane is a two dimensional surface which is spanned by two linearly independent vectors.

Q: Find an equation of the plane. the plane through the point (5, 6, 5) and with normal vector 6i + 8j…

A: As we know that the equation of the plane which is passing through a point Px0,y0,z0  and normal…

Q: Find an equation of the plane that passes through the given point and is perpendicular to the given…

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Q: The equation of the plane passing through the point A(8, 10, 9) and orthogonal to the line x = 7+…

A: Given: Point A(8,10,9) Line x=7+8t , y=6+4t , z=5+9t

Q: Find the equation of a plane that is perpendicular to the vector 3 i - 4j+ 5k and passing through…

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Q: Find an equation of the plane that passes through the given point and is perpendicular to the given…

A: point = (3,5,-7)n=0 i +1 j + 0 kCartesian equation of plane is Ax+By+Cz+D=0(A,B,C)=( 0,1,0)(x,y,z) =…

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: Show that the vector v = [a, b, c]* is perpendicular to the plane P with equation ax + by + cz + d =…

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Q: d. Find an equation (in point-normal form) of the plane containing the points K, M and N Let R be…

A: This is a question from analytical geometry, 3D geometry, and vector algebra.

Q: Find the equation of a plane that is perpendicular to the vector i+ 2j+5 k and passing through the…

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Q: Find the equation for the plane through the point Po = (7,2,3) and normal to the vector n= 4i + 6j +…

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Q: Determine the Cartesian equation of a plane that has x-, y-, and z- intercepts 1/3, 1, and -1/2,…

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Q: snip1

A: we have to find the normal and general forms of the equation of the plane that contains the point…

Q: Determine a unit vector perpendicular to the plane a =21-6j –3k and b=4i+3j – *

A: Since you have asked multiple questions in a single request, we would be answering only the first…

Q: The planes I. and II, have equations IL : 6y+5z =-19, 5 II,: -x -4y+7z =1. State the normal vector…

A: We have to find the normal vectors of the given planes.

Q: Find equation of Plane that an contains the vector: and 1-6 and Z=36

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Q: Find constants a, b, c such that  G( u, v) = ( u + av, bu + v, 2u - c) parametrizes the plane 3x - 4…

A: Given G( u, v) = ( u + av, bu + v, 2u - c) parametrizes the plane 3x - 4 y + z = 5.

Q: Find a vector equation and a parametric equation for the line that passes through the point (5, 1,…

A: We have to solve given problem:

Q: Find an equation of the plane satisfying the given conditions.

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Q: Find an equation of the plane that passes through the given point and is perpendicular to the given…

A: Equation of line passing through the point (x1,y1,z1) and is perpendicular to the plane (ai+bj+ck)…

Q: Find an equation of the plane that contains the point (-1, 2, 3) and is perpendicular to the line…

A: The plane is perpendicular to the line, then the direction vector of the line is normal to the…

Q: Find the cartesian equation of the plane determined by the two intersecting lines x-4 y+1 z-2 L1: 3…

A: Here,  v1=3i+2j+kv2=5i+3j+2k v1×v2=ijk321532=i4-3-j6-5+k9-10=i-j-k Thus, the equation of the plane…

Q: Consider the planes given by the equations x−3y−2z=1, 4x−y+3z=8      Find a vector ' v ' parallel…

A: We have given the equation of plane-                      x−3y−2z=1,                     4x−y+3z=8.…

Q: Is it possible to find a plane perpendicular to both i and j? Explain why.

A: The objective is to find a plane perpendicular to both i and j, if possible.

Q: Use the component form to generate an equation for the plane through P1(4, 1, 5) normal to n1 = i -…

A: Concept - If a plane is passing through (x0, y0, z0) and normal vector is A, B, C.Then equation of…

Q: find the two-dimensional vector equation of the line r(t) = a +t*m such that - r(0) has norm 8 and…

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Q: Q1:A-Find the equation of plane passes through (0,2,-1) parallel to the plane 3x+y+z=7? b- Find…

A: 1. A. According to question,Given plane,3x+y+z=7,Given Point 0, 2, -1

Q: Find an equation of the plane that passes through the given point and is perpendicular to the given…

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Q: e part of the plane with vector equation r(u, v) = (4 + v, u – 3v

A: Here we have to find the area of the part of the plane with vector equation r(u,v)=4+v, u-3v,5-4u+v…

Q: Find the general form of the equation of the plane passing through the point and normal to the…

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Q: Let u = 2 and v = | 0 , and let P = span{u, v}. Find an equation for the plane P.

A: Given: Let xyz be a point in Span {u, v} so there are t, r∈ℝ such that xyz=tu+rv. So,…

Q: Write the parametric and scalar equations of the plane given its vector equation [x, y, z] = [4, 3,…

A: Parametric equation is simple: x = 4 + 2s + 6t y = 3 + s + 6t z = -5 + 4s - 3t

Q: Determine two unit vectors, vand wl, perpendicular to the plane passing through the z-axis and point…

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Q: Let M be a plane that contains the point with coordinates (-2,-2,-1) and is orthogonal to the vector…

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Q: Find the vector equation for the line of intersection of the planes x – 5y + 2z = 1 and æ + z = 1 |…

A: To find the vector equation of the line of intersection of given planes.

Q: Find an equation of the plane passing through the point perpendicular to the given vector Point…

A: Here we have to find the equation of the plane passing through the point (0,4,0) and perpendicular…

Q: (3) Write the equation of the plane which is perpendicular to the vector ñ² = (2₁-1₁-2) and touches…

A: To find- Write the equation of the plane which is perpendicular to the vector n→ = 2, -1, -2 and…

Q: ind an equation for the plane that passes through the point A(1,2,3)parallel to u = 2i + 3 j + k nd…

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