Where does the parametric line (2 + 2t, t, t)intersect the plane x + y – 3z = 4? A) (4, 1, 1) B)1 C) The line and the plane do not intersect, in fact.

Q: Find a set of parametric equations for the line passing through the point (2, 0, 9) that is…

A: a set of parametric equations for the line passing through the point (2, 0, 9)  that is parallel to…

Q: Find the real number a such that the line of x = 1, y = 2-1, z=3+t, the plane of equation parametric…

A: Given parametric equations are x=t , y=2-t, z=3+t Where t∈ℝ and  Plane is αx+5y+z-10=0

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

A:

Q: Find the angle between the two planes 6x − 3y + z = 5,  −x + y + 5z = 5 and find a set of parametric…

A: We have given the equation of planes; 6x − 3y + z = 5 −x + y + 5z = 5 Hence the normal vectors of…

Q: Find the coordinates of the point (x, y, z) on the plane z = 3 x + 3 y + 2 which is closest to the…

A:

Q: Consider the planes II : 3z – y+ 2z=1, Il2 : -9z + 3y – 6z = 24 and the line of parametric equations…

A:

Q: What is the parametric equation for the line of intersection of the planes x + y - z = 2 and 3x - 4y…

A:

Q: Find the symmetrical point of point P(-1,-3,5) with respect to the plane D:3x-2y+z+6=0

A: Point P(-1,-3,5)  plane D:3x-2y+z+6=0

Q: Let M be a plane with the following properties: • M contains the point with coordinates (0,0,1) • M…

A:

Q: Give a scalar equation for a plane at distance 9 from the plane P with equation -2x-2y+z = 4.

A:

Q: The vector equation r(u,v)=ucosvi+usinvj+vk, 0≤v≤4π, 0≤u≤1, describes a helicoid (spiral ramp). What…

A:

Q: Let L be the line with parametric equations x = 2+t, y = 5-2t, z = 4 + 3t. Let S be the plane that…

A: Given L is the line with parametric equations x = 2 + t , y = 5-2t , z = 4 + 3t . Let S be the plane…

Q: The vector (x, 2y, –3z) is normal to the surface x2 + 2y? – 32 – 4 = 0 at any point (x, y, z). - - -

A: Let's find.

Q: Find the angle between the two planes 3x + 2y − z = 7,  x − 4y + 2z = 0 and find a set of parametric…

A: Known fact: The angle between two planes a1x+b1y+c1z+d1=0  and a2x+b2y+c2z+d2=0 is given by…

Q: Find a parametric equation for the curve of intersection of the cone x²+y² = z² and the plane x+2z =…

A: We have to find a parametric equation for the curve of intersection of the cone x2+y2=z2 and the…

Q: = xyi + yzĵ + de is a in x-y plane.

A:

Q: Find the distance between the given point and the given plane. Point: (−1, 4, −2) Plane: x+2y+z=3

A: We can use distance formula to calculate the distance.

Q: 20. Find the coordinates of the point where the line with parametric equations x =1-1, y=3+t, z=3-21…

A: If they intersect the XOZ plane then the value of y coordinate is 0.

Q: Determine the parametric equations of the curve of intersection of the paraboloid x? – y? + z² = 0…

A: Detailed explanation mentioned below

Q: Calculate fc((3x + 5y) i + (3x + 4y) j) - dr where C is the circular path with center (a, b) and…

A: According to the greens theorem:

Q: The planes 5x+y+z=−6 and 3y+5z=22 are not parallel, so they must intersect along a line that is…

A: Given planes are 5x+y+z=-6 and 3y+5z=22 find the vector parametric equation

Q: Find the angle between the two planes −2x + y + z = 2,  6x − 3y + 2z = 4 and find a set of…

A: The given planes are −2x + y + z = 2  and  6x − 3y + 2z = 4 . Obtain the angle between them as…

Q: What is the parametric equation of line through the point (1,2,-1) and perpendicular to the plane…

A:

Q: What are the parametric equations for the intersection of the planes x - y - z = 1 and 2x + 3y + z…

A:

Q: Find a scalar equation of the plane that contains the linex = 1 + t, y = 2 − t, z = 4 − 3tand is…

A: Given,The plane that contains the line x = 1 + t, y = 2 − t, z = 4 − 3t.So, we can say that the…

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: Find the angle between the two planes 3x − y + z = 7 ,  4x + 6y + 3z = 2 and find a set of…

A: Given, two planes 3x − y + z = 7 ,  4x + 6y + 3z = 2  we have to find the angle between the two…

Q: Find a parametric representation for the surface: The plane through the origin that contains the…

A:

Q: Which of these describes a parametrized line perpendicular to the plane 3x + 2y + z = 15? A) (2 +…

A: The direction ratio of the normal vector to the plane is (3, 2,1 ) So the perpendicular line to the…

Q: Write down a parametrization r(u, v) for the portion of the plane x - y+ z = 3 that lies inside the…

A: r(u, v) = ( u cos v , u sin v,  3 - u cos v + u sin v)  Where u € [0, 1]           v€ [ 0,2π]

Q: Provide the point-normal form equation of the plane containing (1, -1, 4) and line with parametric…

A:

Q: ). Find a parametric representation for the line of intersection of the following two planes: I - y…

A: We use cross product to find the direct vector for the line.

Q: Let P(r, y, z) be the point at which the line with parametric equations x = -t, y=3+2t, z=1-t te R,…

A:

Q: Show that the curve al lies in plane and b) finel the equation of the plane that contains the curve…

A:

Q: What is the parametric equation for the line through the point (1,2, 3) paralkel to the y- axis? (a)…

A:

Q: Sketch or describe the surfaces in R³ determined by the equa- tions in Exercises 40–46. z = y? + 2

A:

Q: Find the value of d in the equation of the plane ax+by+cz+d=0 passing through the point P(3, 1, -2)…

A:

Q: Find feF- di utere Cisa crce of radus 2 in the plane z+y+ = = 9, centered at (2, 2, 5) and oriented…

A: The given function is, F→=2yi→-2xj→+3y-zk→ So, the curl of can be calculated as, curl…

Q: Bounded by the paraboloid z = 10 – 3x? – 3y2 and the plane z = 4 (using polar coordinates)

A:

Q: Flnd the parametric equation of the line of ntersection of the planes and

A: First find normal vector then take cross product and use formula

Q: Find a set of parametric equations for the line passing through the point       (1, 0, 2) that is…

A:

Q: Find the point at which the line with parametric equations x = 3 + 4t, y = 5 – 2t, z = 4 + 7t…

A: According to question,Given equation of line in parametric form,x = 3+4t,y = 5-2t,z= 4 +7t,Equation…

Q: Find the real number a such that the line of x = 1, y = 2 – 1, z = 3 +t, parametric equations te R…

A: Given, parametric equations are                               x = t, y = 2-t, z = 3 +t, t in R To…

Q: Find a point-normal form equation of the plane containing (1, -1, 4) and line with parametric…

A:

Q: Derive the parametric equations of the line of intersection of the two planes: II : x – 2y + z =1…

A: Given: P1:   x−2y+z=1P2:   2x−2y−4z=4

Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.