Find the equation of the plane containing P and orthogonal to the line r(t). Write your answer in the form ax + by + cz = d. Make sure to justify your choice of normal vector using a diagram and/or words.

Q: Find a unit vector that is parallel to the line tangent to the parabola y = x2 at the point (6, 36).

A: The parabola y = x2 and the point (6, 36)

Q: Find a vector normal to the plane with the given equation.9x − 4y − 11z = 2

A: Our Aim is to find a vector normal to the plane with the equation given below:-9x-4y-11x=2 -(i)

Q: Find the new equation of the line y=x after applying the translation by the vector (2, -1)

A: New line will be shifted to 2 units right and 1 unit down.   Point on x axis (2,0) Point on y axis…

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

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

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Q: Find the vector equation for the line of intersection of the planes 5x +y-3z= -3 and 5x + z = -1 r =…

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

A: Topic = vector

Q: Given the points A(2,6,0) and B(3,5,6) determine the equation of a line vector in parametrix form

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Q: write the equation of the line passing through P with direction vector d in (a) vector form and (b)…

A: Given that : P = ( 1, 0 ) and d = -13 .

Q: Find parametric equations for the lines in The line through the origin parallel to the vector 2j + k

A: Given  The vector equation is  0i +2j+k  The line passing through origin , origin is  ( 0 ,0 ,0)…

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

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Q: point for the vect

A: What point is the endpoint for the vector corresponding to T(x)?

Q: For a plane with a normal of n = [1, 2, - 9]   and containing a point P (3, -4, 0), write the…

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

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Q: Given a vector A = 3a, - 2a, - 16a, find its components normal to the plane 2x + 2y – z + 8 = 0 and…

A: We have to determine component normal to the plane

Q: Let L be the line defined by the vector equation v = [ -1, -3 ] + t [ 1, -2 ]. Find the equation y =…

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Q: Find the equation for the plane through the point Po = (3,4,3) and normal to the vector n=21 + 7j +…

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Q: Find the cartesian equation of the plane that is normal to the vector n = -3i – 4j+4k and includes…

A: n=-3i-4j+4k  and P(2,-7,5)

Q: Find the equation of a line in vector form that is parallel to 3x−2y+z= 4, passes through the point…

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

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Q: Find a vector equation for the line segment from (2, -4, 5) to (6, 4, 4). (Use the parameter t.)…

A: Explanation of the solution is given below...

Q: Find equation of Plane that an contains the vector: and 1-6 and Z=36

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

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Q: Give a vector parametric equation for the line through the point (4, 2) that is perpendicular to the…

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Q: Find the coordinates of the point P on the line and a vector v parallel to the line. x = 8t, y = 8 –…

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Q: In vector form, the equation of a line is given by r = (0, 1, 2) + t(2, 1,0), where t e R. T…

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Q: write the equation of the line passing through P with direction vector d in (a) vector form and (b)…

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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: For the given vector function below, find the equation of the plane that contains the osculating…

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Q: Write the line through p = (3, 2, 2) in the direction of v = (-1, –4, 2) in vector parametric form:…

A: We have to write the line trough p=3,2,2 in the direction of v¯=-1,-4,2 in vector parametric form.

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: 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: Give a vector parametric equation for the line through the point (-1, –1) that is perpendicular to…

A: We have to write a parametric equation of the line  which is passing through the point (-1, -1)…

Q: The unit vector normal to the plane 2x + 3y + 6z = 7 is

A: The given plane is 2x + 3y + 6z=7. The unit normal vector to fx,y is defined as n=∇F∇F. The normal…

Q: Show that A(x - x0) + B( y - y0) = 0 is an equation for the line in the xy-plane through the point…

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

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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)…

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Q: Give a vector parametric equation for the line through the point (1,−4,1) that is parallel to the…

A: We have to find the vector parametric equation for the line through the point (1,-4,1) that is…

Q: Find an equation of the plane that passes through the given point and is perpendicular to the given…

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Q: Sketch the curve whose vector equation is r(t) =

A: given rt=<tcost,tsin,t>

Q: Find a vector equation and parametric equations for the line segment that joins P to Q, given P(0,…

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Q: Sketch the curve with the vector equation 7(t) = (t,2 – t, 2t). Indicate with an arrow the direction…

A: Solution

Q: Find the coordinates of the point P on the line and a vector v parallel to the line. x = 6t, y = 7 –…

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Q: The surface of a hill can be given as h(x, y) = e-(z²+y²), If a ball is placed at the (2, y)…

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