Consider the line with vector equation r(t) = (2t, t+1, t) and the point (1, –3, 2). a. Find the equation of the plane that contains both the given line and the given point. b. Find the equation of the plane that contains the given point and is perpendicular to the given line.

Q: Is the line x = 9- 8t, y= 4 + 2t, z= - 5t parallel to the plane 2x + y-z=2? Give reasons for your…

A:

Q: Find an equation for the plane that contains the line v = (-1, 1, 2) + t(9, 5, 4) and is…

A:

Q: Find the Cartesian equation of the plane containing the point P(1,3,-2) and the I T = (0, -1, 4) +…

A:

Q: Find a vector parallel to the line of intersection of the planes 2x + 3y – = =11 and 4x + 2y – 3= =…

A: Topic:- vector

Q: Find a vector parametrization of the line through P = (3, –8, 5) in the direction v = (-2, 0, 3)…

A:

Q: Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t…

A: Topic = vector

Q: If a line is parallel to the vector 2i + 7j + 10k and passes through the point P = (6, 10, 5), then…

A:

Q: Find the vector equation for the line of intersection of the planes x + 4y + z = -5 and x + 4z = 0…

A: Here we have to find the vector equation for the line of intersection of the planes x+4y+z = -5 and…

Q: Find the coordinates of the point P on the line and a vector v parallel to the line. X = 4t, y = 4 –…

A:

Q: Consider the planes x – 10y – 2z = 3 and 2x – y + z = -13. Find the following. 1. A vector…

A: The given planes are: x-10y-2z=3 and 2x-y+z=-13 Let us assume z=t on the line of intersection, where…

Q: The curve with vector equation r(t) = t'i + 2tj+ 3tk is %3D a line.

A:

Q: Find the vector equation of the line of intersection of the planes x+y+z=1 and x+2y+2z=1.

A:

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: 14. Given the vector form of a line in three-space to be [x, y, z] = [2,6, – 3] + t[4,10,6], find a.…

A:

Q: Show that the curve a(t) = (t,4,²) lies in a plane and find the equation of this plane.

A: Solution:- αt=t,1+tt,1-t2t t,1t+1,1t-t=x,y,z It is clear that, t-1t-1+1t-t=x-y+z x-y+z=-1

Q: If possible, determine the vector, parametric,symmetrical and reduced line r that: a) passes…

A: Given the lines S1:t·1,2,3, t∈ℝ and S2:y=x+2, z=x. To find a line L passing through the point P1,1,1…

Q: Find parametric equations for the line whose vector equation is given. (а) (т, у) 3 (6, —5) + t(1,…

A:

Q: Find the vector equation for the line of intersection of the planes 32 + 2y :0 ánd 3* + ,0 ) +t(6, r…

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

Q: Find a vector normal to the plane with the given equation. 2x + 8y + 9z = 3 n =

A:

Q: (a) Find the y-intercept of the line in 2- space that is represented by the vector equation r = (3 +…

A: Given : a) The line in 2- space that is represented by the vector equation r = (3 + 2t)i + 5tj .

Q: Find the coordinates of the point P on the line and a vector v parallel to the line. x = 7t, y = 4 -…

A: Given, The equation of the line x = 7t, y = 4-t, z = 3+9t We have to find the coordinate of the…

Q: Find a vector normal to the plane -2x - 3y = 12 - 4z.

A: Given  a vector -2x-y=12-4z. We need to find the vector normal to the plane.

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

A: Given  equation of planes 2x-2y-5z=5 2x+2z=1

Q: Find the general form of the equation of the plane containing the point Q(0, –2, 10) and the lin l1…

A: Let's find.

Q: Find the vector equation that represents the curve of intersection of the paraboloid z 2x2 + y and…

A: z=2x2+y2  & x=ey We can see that y can be represented in terms of x.

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: (c) Find the vector equation of the line of intersection of the planes x-2y+z=0 and x+y=z= 1.

A:

Q: Find the coordinates of the point P on the line and a vector v parallel to the line. x = 3t, y =…

A:

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: Find the vector form of the equation of the line inR3 that passes through P = ( - 1, 0, 3) and is…

A: Find the vector form of equation of line passing through point  P=-1,0,3 and perpendicular to the…

Q: Find the coordinates of the point P on the line and a vector v parallel to the line x = 3t, y = 1 –…

A: Given parametric equation of line x=3t, y=1-t, z=3+5t

Q: Let r(t)=(t, (4/3)t^(3/2), t^2). Find the equation of the osculating plane at the point…

A: Let the given point be (x0, y0, z0), to find the osculating plane find the cross product of the…

Q: Consider the line l in R³ given by the parametric vector equation 3 1 -2 -1 and the plane described…

A: Given that the line l in R3 given by the parametric vector equation and the plane described by the…

Q: Find a vector parametric equation (t) for the line through the points P = (1, −4) and Q = (−2, −9).

A:

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

A: Given:u = 2i + 3j + k and v = i - j + 2kFirst find cross product

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…

A:

Q: Find the vector equation of the line between the points (3, -1, 2) & (1, 4, -1), then graph the…

A:

Q: Find a vector equation for the line through the point P = (4, -2, -4) and parallel to the vector vv…

A: P= (4,-2,-4) is a point on the line.  And the line is parallel to v=(4,-1,4)  v is the velocity…

Q: Find the vector equation of the line of intersection of the planes x+y+z = 1 and x+2y + 2z = 1.

A: The equation of the line vector of the intersection of 2 planes can be calculated using the cross…

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

A:

Q: Find parametric equations for the line whose vector equation is (a) (x, y) = (3, –5) + t(1, –2) X =…

A:

Q: c) Find an equation of the plane that contains the line L: 1=1-t, y = 2+t, z = -3t where teR and the…

A: Equation of plane that contains the line x=1-t, y=2+t, z=-3t where t is in R and the point…

Q: Find parametric equations for the line whose vector equation is given. (а) («, у) %3D (3, —3) + t(1,…

A:

Q: Find the vector equation that represents the curve of intersection of the paraboloid z = 4x2 + y and…

A: The given curves are: z=4x2+y2     and     x=ey

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: Let L be the line passing through the point with coordinates (6,  6, 8) in the direction of the…

A:

Q: 14. Given the vector form of a line in three-space to be [x, y, z] = [2,6, – 3] + t[4,10,6], find a.…

A:

Q: 2. Find the equation of plane which passes through the point (1,2, –3) and the vector i – 2j + 3k is…

A: