16 12 25 25 9 12 25- .A = (projection onto the line through the 25 origin with direction vector

Q: 2. Let f(x,y) = t Find the direction of a unit vector so that Daf is minimal at point (0, 1). %3D 3…

A: To find the direction of a unit vector u so that the directional derivative of f at point (0,1) in…

Q: write the equation of the plane with normal vector n passing through the given point in the scalar…

A: We have, Equation of plane through P0 =(x0, y0, z0) with normal vector n = <a, b, c> Vector…

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: Write the line through p=(5,2,2) in the direction of v¯=⟨3,5,0⟩ in vector parametric form: L(t) =…

A: We know that the equation of line passing through the point P in the direction of v→ is…

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: Consider the vectors a = (-1,–2, 3), b = (3,0, –2), v = (2, –5, –4) and the plane P given by 2x + y…

A: The given vector is; a→=-1,-2,3b→=3,0,-2v→=⟨2,-5,-4⟩and Plane P is;2x+y=z+13⇒2x+y-z=13…

Q: Which of the following vectors is orthogonal to the plane 3(x - 7) + (y + 8) - 5z = 15? O O O

A: Given equations of plane 3(x-7) +(y+8)-5z=15this is an equation of the form a(x-x1)+b(y-y1)+…

Q: Let a and b be two vectors. Prove that (b – proja (b)) · a = 0. Hint: Use the definition of the…

A:

Q: The vector projection of u = [3, -4, -2] on the x-axis is: %3D 3 29 3 29 3í

A: The direction ratios of the X-axis are, (1,0,0) Also, u•(1,0,0) = 3 is the scalar projection on the…

Q: 11. Use vectors to show that the distance from P:(x1, yı) to the line ax + by = c is |ax, + by, – c|…

A:

Q: Find parametric equations for the line through the origin parallel to the vector 3j + 9k. Let z =…

A:

Q: Q2/ Show that the vector v = ai + bj is perpendicular to the line ах + by + с.

A: A line and the vector  are said to be perpendicular to each other when the product of their sloped…

Q: 13. Consider the position vector i = (1,0, 0) in R³. Describe the set of position vectors v in R³…

A: Given, i=1,0,0∈ℝ3

Q: Find a vector function that represents the curve of intersection of two surfaces: z = x² - y? and x²…

A: Idea  1) Solve the equation in term of only one variable (2) assume in term of parametric function…

Q: 6. Find the distance of the point (-1,1) to the line given in vector form (x, y) = (2,0) +t(4, 3).

A: Given that: The point is (-1, 1).The vector is:  (x, y)=(2, 0)+t(4, 3)

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: (a) Let ri(t) = [2, 1, – 1] + t[1, 1, 1] be the equation of a line L1 in R³. De- termine the…

A:

Q: Is it true or false that the vector u = (3, 3, –3) is a direction vector to the line x + y+ 2z = 6 l…

A: I have written equation of line in parametric form and then checked the direction ratio

Q: 2. If X(-2, -6, 5) and Y(3, -4, 1) are two points in R³ , determine: a) XY b) | XY | in reduced…

A:

Q: Represent the line segment from P to Q by a vector-valued function and by a set of parametric…

A: Two vectors Px1,y1,z1 and Qx2,y2,z2 Line segment P to Q: 1) Parametric form is…

Q: Let v = ( 1, 2 ), Find a vector w with the same direction as v such that || w || = 7.

A: Let v=1,2 Then,  v=12+22=5 The unit vector in the direction of v is, v=vv=1,25=15,25

Q: Prove that the vector T′(t) is 0 for an object moving in a straight line.

A: Here an object moving in a straight line. The path of the object can be given as follows.…

Q: Q1. If O (0,0,0) in (x,y,z) 3-D space and r(2,5,1) draw the vector in 3D and find its direction.

A: Plot point in 3D

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: Find a vector orthogonal to the line x+y=1

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: The vector is orthogonal to the plane 4x + 5y –7z =1 Select one: O True O False

A:

Q: c) Show that vector ū = 1 cannot be expressed as a linear combination of 2 in R3 3 vectors vi 1 and…

A: we have to show that , we can not write as a linear combination of vectors v1 and v2

Q: Show that direction vectors of the lines x-3y+ 1 L: and 2 -2 are orthogonal.

A: Direction vector of a line x-x1a=y-y1b=z-z1c   is   <a,b,c>  So, Write the direction vector of…

Q: Consider the vector y = [1 2 3]. Find the orthogonal projection of y onto the plane x, - 2x2 + 4x3 =…

A: Given vector is y=1,2,3 Given plane is x1-2x2+4x3=0 Normal vector to the p[lane is n=1,-2,4…

Q: Find a vector tangent to the curve of intersection of the two cyclinders ? + y² = 98 and y² + z² =…

A:

Q: Find a vector parametrization of the line through P = (0, –8, 7) in the direction v = (7, 0, 0)

A:

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

A:

Q: Which vector is the orthogonal projection of v = 0 onto W = span -2 °() V3 V6 3 V6

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: Find a non-zero vector 7 perpendicular to the vector u = 5= [8]. 6

A:

Q: Find a nonzero vector parallel to the line of intersection of the two planes 5y – 4x + z =-1 and –…

A: The line of intersection of two planes is orthogonal to their respective normals.

Q: Consider a point Q(x1, y1, z1) and a line L given by the vector equation r = r0 +tv,where r0 =…

A: First obtain a, where a is the distance between R0 and Q.

Q: 1. Show that A(r - ro) + B(y - yo) =0 is an equation for the line in the ry-plane through the point…

A:

Q: . Let 7 - " be the separation vector from a fixed point (a', y', z') to the point (x, y, z). Show…

A: we have,  fixed point x',y',z'=r' and point x,y,z =r  r→-r→'=x-x'i^+y-y'j^+z-z'k^ a…

Q: generate a parametrization of the line through P(2, -4, 7) parallel to v1 = 2i - j + 3k. Then…

A: We know that equation of line through (x1,y1,z1)  and parallel to ai+bj+ck is

Q: Find an equation of the plane that passes through the point P(3, 5, 1) and has the vector n = (1, 8,…

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

Q: 32. a) Determine the vector equation of the line of intersection of these 2 planes: x + 2y + 2z = -3…

A: According to Bartleby guidelines I can only answer the first question of the given two problems.

Q: 12. Suppose that ú = (a, b) is a non-zero vector in R², where a ± 0. Show that the line L =…

A:

Q: Consider the vectors H [0] H . If I is the line defined by the vector w₁, fi W1 = [o] W2

A: Here, in the question three vectors are given:  w1, w2, b  L is the line defined by the vector w1 We…

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

A: Answer

Q: 2. Determine the vector equation of the plane that contains the following two lines. L1: 7 = [ 4,…

A: