Fix one vertex at (0,a) in the (x,y)-plane and move another vertex along the x-axis. Show that the locus of the third vertex (x,y) of an equilateral triangle consists of 2 lines, y + a = ±v3x

Q: B. Show that line is parallel to plane x-2y + z 6

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Q: Find the point at which the line T(t) = (4, 1, 7) +t( – 5, – 7, – 4) intersects the yz plane.

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Q: The point of intersection between the line t with the plane 2x + 3y +5z = 31 is

A: Given line ,x-1-4=y-13=z5=tTherefore any point on the line is of the form (1-4t,1+3t,5t)Put this…

Q: Find the points in which the line x = 1 + 2t, y =-1 - t, z = 3t meets the three coordinate planes.

A: Given line is  x = 1 + 2t, y =-1 - t, z = 3tWhen the line meets xy plane then z=0So, 3t=0Or, t=0

Q: What angle does the line of intersection of the planes 2x + y - z = 0 and x + y + 2z = 0 make with…

A: Given:The planes 2x + y - z = 0 and x + y + 2z = 0

Q: Find an equation for the set of points equidistant from the xy-plane and the yz-plane.

A: Find an equation for the set of points equidistant from the xy-plane and the yz-plane.

Q: Find an equation for the set of points equidistant from the a. xy-plane and yz-plane. b. x-axis and…

A: a) Let (x,y,z) be such point that xy-plane and yz-plane are equidistant from it.Distance of (x,y,z)…

Q: Find the minimum distance from the point (8, 3, 4) to the plane x - y + z = 4. (Hint: To simplify…

A: Let ax+by+cz+d=0 is a plane  Then distance from the  a point (xo,yo,zo) is D= |axo +byo +czo…

Q: 1. Find the distance from the point S(2,-3,4) to the plane x+2y+ 2z =13

A: "Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Find the coordinates of the point located in the xy-plane, five units to the right of the xz-plane,…

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Q: Find the​ distance, d, between the point S(6,9,10) and the plane 10x+7y+1z=60.

A: Answer is 73/5√6.

Q: Determine the location of a point (x, y, z) that satisfies the condition y < 0

A: With the given condition there are 4 scenarios 1) x>0,y<0,z>0 2) x<0,y<0,z<0 3)…

Q: Find the minimum distance from the point (6, 0, 9) to the plane x – y + z = 5. (Hint: To simplify…

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Q: Find the shortest distance, d, from the point (4, 0, –4) to the plane x + y + z = 4.

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Q: Find all hyperbolic lines that passes through i +1 and parallel to the y-axis.

A: Find all hyperbolic lines that passes through i+1 and parallelto the y-axis.

Q: What angle does the line of interaction of the planes 2x+y-z = 0 and x+y+2z= 0 make with the…

A: The direction vector of the line of the interaction of the two planes is equal to the cross product…

Q: Find the distance between the parallel planes. x + y + z = 1 and x + y + z = 3

A: The distance between two parallel planes is the shortest distance between their surfaces. Distance…

Q: A sphere of constant radius r passes through the origin O, and cuts the axes in A, B, C. Prove that…

A: Given: Equation of the sphere with origin at O x2+y2+z2-ax-by-cz=0→1The radius of the sphere is…

Q: Find the shortest distance from the point (2,0, –3) to the plane x + y + z = 1.

A: This is a problem related to geometry part. Based on the concept of plane and point we will find the…

Q: Find the point P nearest to Q = (−1, 3, −1) on the planex − 4z = 2

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Q: Find the shortest distance, d, from the point (6, 0, -4) to the plane x + y + z = 4.

A: The shortest distance of a point from a plane is said to be along the line perpendicular to the…

Q: Find the equation of the plane which passes through O and is parallel (5а + бу) — 3 to z – %3D

A: The equation of given plane is,      z - (5x + 6y) = 3⇒ z - 5x - 6y = d   for a plane of the form ax…

Q: Find the shortest distance between the point A(2, 5, –7) and the plane 3x – 5y + 6z = 9.

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: If the line L: X = 3 + 2t , Y= 2t , Z=t intersects the plane X+ 3y Z -4 in a point P. The…

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Q: Find the coordinates of the point located in the xy-plane, four units to the right of the xz-plane,…

A: Let the coordinate of the point located in the xy-plane is x,y,0

Q: Find the point of intersection of the line x = 2 − 3t y = 4 + 2t −∞ < t < ∞, z = 3 − 5t, and…

A: x=2-3ty=4+2tz=3-5tand the plane 2x+3y+4z+8=0

Q: Find equation of Plane teter mined by the Lines Li. X = t-29 y = 2t+3 Z= +4 and Lz! X…

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Q: 8) Determine if the line given by r(t) = intersects the xz-plane. If so, determine at what point.

A: we have to find the point of intersection.

Q: Find the minimum distance from the point (6, 0, 5) to the plane x - y + z = 3. (Hint: To simplify…

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Q: Find the distance between the given parallel planes. 2x - 3y + z = 8, 4x - 6y + 2z = 1

A: The solution are next step

Q: Draw a diagram showing two perpendicular lines that intersecton the -axis and are both tangent to…

A: The given parabola equation , y=x2. To draw two perpendicular lines that intersect on the axis and…

Q: Find the shortest distance from the point (0,2,3) to the plane x+2y+z=1

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Q: Show that the line in which the planes x + 2y - 2z = 5 and 5x - 2y - z = 0 intersect is parallel to…

A: Find the vectors normal to the planes x + 2y - 2z = 5 and 5x - 2y - z = 0.

Q: Find the shortest distance between (3,4,1 ) and the plane X+y +Z = -2

A: Using distance between plane and point formula we can calculate.

Q: Find the distance from S(1, 1, 3) to the plane 3x + 2y + 6z = 6.

A: It is given that a point S1, 1, 3 and the equation of the plane 3x+2y+6z=6   Here x1=1, y1=1, z1=3…

Q: -6x - Find dA, where Ris the parallelogram enclosed by the lines -3x+y R – 6x – y = 0, – 6x – y = 2,…

A: I have used jacobian to solve the area of above double integral.thanks

Q: Find the shortest distance from the origin to the line of intersection of the planes 2x + y − z = 1…

A: Given: The planes: 2x+y-z=1 and x-y+z=2.

Q: Find the acute angle (in degrees) between the planes 5x + y + 4z = O and a y + 2z = – 3.

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Q: Show that the point (x0 , y0) closest to the origin on the line ax+ by =c has coordinates

A: Distance between x0, y0 and origin (0, 0) is, D=x0-02+y0-02D=x02+y02D2=x02+y02f=x02+y02…

Q: Find the points of intersection of the line x = 3+2t, y =5+7t, z = -3+t, that is, I(t) = 3+2t, 5+7t,…

A: To determine the intersection of the line with the coordinate planes.

Q: Find the angle between the planes (x+y+za1)and. (x-24+32-1). And find the Symmetricequations for the…

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Q: Find the point at which the line intersects the given plane. x = 3 - 3t, y = 4t, z = 1 + t; x + 2y -…

A: Given line ,x=3-3ty=4tz=1+t   -1and plane , x+2y-z=6

Q: Find the point at which the line intersects the given plane. x = 4 - t, y = 5 + t, z = 3t; x - y +…

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Q: Show that line y+1 is parallel to plane x- 2y + z = 6 С.

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Q: Find the points PCo10,0),Q Ce,2,0) and RCOl- thal- 2I to the line: the Line in the Plane of passes…

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Q: Find the shortest distance from the point (1, 0, −9) to the plane x + 2y + z = 9.

A: Given plane isx + 2y + z = 9and point (1, 0, -9)We have to find shortest distance

Q: Find the point of intersection between the planes TT :x+y-z =1, n, : 2x – 3y -9z = 10 and n, : x +…

A: The three planes are, π1=x+y-z=1 π2=2x-3y-9z=10 π3=x+2y-3z=-4

Q: Find the distance from the point to the plane. (3, 0, 10), 2.x + 3y + z = 2

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