(a) For a given vector a 7 0, describe the loci (the set of points 7 = determined by the equations: (x, y, z)) on R³ (i) d (2-2, у+1, 2—3) — 0. (ii) ||7 – a|| = 21. %3D

Q: The figure shows a vector a in the xy-plane and a vector b in the direction of k. Their lengths are…

A: →A vector along z-axis is perpendicular to any vector lying in xy-plane.→a→×b→ is perpendicular to…

Q: P that is orthogonal to the vector AB. . B e distance from the point P(-3,1,9) В(-2,3, —4).

A:

Q: If the vector K ag+Lay+ Ma, shows the maximum increase rate and direction of the scalar Vat point…

A: If the vector Ka^x+La^y+Ma^z shows the maximum increase rate and direction of the scalar V at point…

Q: Let L be the line through the point Q=(−3, 3, −2) and direction vector d=[4, 5, 1]T. Find scalar…

A: We need to find equation of planes.

Q: (a) Find a vector parallel to the line of intersection of the planes 4x – y – 3z = 6 and -5x + 4y –…

A:

Q: A parametric equation of the line (D) passing through A and parallel to the vector v is: x= 2+3t y…

A:

Q: 2. The plane containing the point P(Xg» Yor Zo) and having the normal vector n = (a, b, c) is…

A: Equation of a plane passing through the point Px0, y0, z0 and perpendicular to the vector n=a, b, c…

Q: Find a vector n normal to the plane with the equation 5x - 5y – z = 10. n = (5,

A: Given equation of plane is 5x-5y-z=-10.want to find:Normal vector n to given plane.

Q: (b) Suppose that the position vector of a particle moving in the plane is r-12vii+j, t>0. Find the…

A: This is application of derivative problem

Q: Find a vector of magnitude 2 parallel to the line of intersection of the planes x + 2y + z - 1 = 0…

A: Find a vector of magnitude 2 parallel to the line of intersection of the planes x + 2y + z - 1 = 0…

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

A:

Q: X, y, z) = (10y - az – 5x)i + (3by-x +8z)k + (z – 5y + 2cx)j (2) Find the real values of a,b,c that…

A:

Q: 3. Find a vector function that represents the curve of intersection of the paraboloid z =. and the…

A: To find the vector function that represents the curve of intersection of the paraboloid z=x2+y2, and…

Q: Find a vector parametrization 7 (t) for the intersection between the two surfaces: x² + 10y4 + 223 =…

A: According to the given information, it is required to find a vector parametrization for the…

Q: Thẻ nòrmal vector to the plane 3x + 4y – 5z – 6 = 0 is A 3i+ 4j – Sk B 3i + 4j + Sk C 3i+ 4j – 6k D…

A: It is normal vector form.

Q: Show that the vector form (x, y, z) = (-5, 1,0) + t(3,2, 1) for some parameter t ER satisfies the…

A:

Q: 10. Draw the graph of the vector-valued functions (a) for the plane curve, r(t) = 2cost³i + 3sint³j,…

A:

Q: The figure shows a vector a in the xy-plane and a vector b in the direction of k. Their lengths are…

A: In the question we have to find cross product and direction.

Q: 3. Write the scalar equation for a plane passing through (5,8,-1) and perpendicular to the line x =…

A:

Q: The vector equation for the curve of intersection of the surfaces x = y² and z = y in terms of the…

A:

Q: Find a vector parametrization r(t) for the line with the given description. Passes through (-2, 0,…

A:

Q: Find the point on the graph of z = 3x2 – 2y? at which vector n = (-12, –16, –2) is normal to the…

A: Shown below

Q: 1. The acceleration of an object is a=2i+4j+6k.  Compute the position vector r(t) if the object at…

A: a=double derivative of position vector

Q: 1. Consider f(x, y) = 2x² – y² + 3x – y and U is any unit vector in R2. Find the maximum value of…

A: To find maximum value of DUf at x=1, y= -2 we shall use the given definition

Q: Which of the following is the equation of the plane passing through the point (0,2,-3) and with…

A:

Q: Suppose that ∇ƒ(x, y, z) is always parallel to the position vector x i + y j + z k. Show that ƒ(0,…

A:

Q: 96. An ant is at the point (1, 1, 3) on the surface of a bowl with equation z = x² + 2y², where x…

A:

Q: 3. Find the length of the trajectory of the vector valued function f(t) = (7V2.1,7",e-7) from 0 to…

A: The detailed solution of the problem is given in the next steps.

Q: 4. A vector normal to the paraboloid z = (x² + y*)/2 is (a) (х, у.1). (b) (2x,2y,–1). (e) (-х,-у.1).

A:

Q: 1. Consider f(x, y) = 2x² – y² +3x– y and U is any unit vector in R?. Find the maximum value of D7f…

A:

Q: 4) a) Find the vector parametric equation of the line in R³ that is perpendicular to the plane 2x -…

A: Given plane is, 2x-3y+7z-4=0

Q: Find the vectors T, N, and B at the given point. r(t) = (6 cos(t), 6sin(t), 6 In(cos(t))), (6, 0, 0)…

A:

Q: Find a vector function that represents the curve of intersection of the paraboloid z=7x^2+5y^2 and…

A:

Q: Which of the following is the equation of the plane passing through the point (0,2,-3) and with…

A:

Q: Match the vector-valued function with its graph. r(u, v) = ui + vj + uvk -2-1 2 2

A:

Q: 7. (a) What is the projection matrix that projects all vectors in R³ onto the plane x+ y + z = 0?…

A:

Q: (2t, 1 – t, 4t) is a parametrization of the line passing through the point (0, 1,0) and also…

A: The statement : <2t, 1-t, 4t> is a parametrization of the line passing through the point (0,…

Q: A body is spiraling upward due to rapidly rising air on a path similar to that of a helix and having…

A: Consider the position vector.

Q: A velocity vector of v = 3t2i − 6t3 j − 4K ft/s  is governing the movement of an object in the 3D…

A: It is given that v=3t2i-6t3j-4kfts. Also, object starts at -1,4,5 at t=0. That is, s0=-i+4j+5k.…

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: Q1) Find equation of plane normal to vector 3i + 2j – k and pass through point (1,1,5).

A:

Q: (b) Let r(t)-ti- +(4-t) k, then the vector equation of the line tangent to the graph of r(e) at the…

A: Given:- r(t) =t2 i -(1+t)-1j +(4-t2)k To find:- Vector Equation of tangent line passing through…

Q: 1) Find a vector function r(t) that represents the intersection of the surfaces: = /x² + y² z = y +…

A:

Q: 2.) a.) y=fcx), X=t y = f (t) the vector egn of the plane curve is ずul-t+fc+)+o.g は13

A:

Q: 9. Find the normal unit vector n to the surface 62 + 2y²+ 2² = 225 at the point P: (5,5, 5).

A:

Q: Find a vector-valued function whose graph is the indicated surface: the part of the paraboloid z =…

A:

Q: Find the scalar equation of the plane that passes through point P(−3, 3, 5) and is perpendicular to…

A: To Find: Scalar equation of the plane that passes through point P(−3, 3, 5) and is perpendicular to…

Q: 4. (8 points) Find a vector parametrization for the intersection of the plane x +y+z = 3 and the…

A: Use polar coordinates for Parameterization