Find a plane containing the point (– 3, – 3, 7(t) = (– 2, – 3, – 5) + t(3, 4) and the line 6, 7).

Q: Find the point at which the line T(t) = (4, 1, 7) +t( – 5, – 7, – 4) intersects the yz plane.

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Q: Find a plane containing the point (7,3,5) and the line x(t) - 3 + t | y(t) 8 +t z(t) - 3 + 5t

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Q: 5. Find the equation of the plane containing the points P(3, 2, 1), Q(2, 1, -1), and R(-1, 3, 2)

A: Let position vectors of point P(3,2,1) , Q(2,1,-1), and R(-1,3,2) are a→,b→ and c→

Q: Find a function r(t) that describes the line through P(3,4,5) that is orthogonal to the plane 2x-z=4

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Q: Find an equation of the plane containing the point (3,0,-4) and perpendicular to the line r(t)=…

A: A detailed solution is given below

Q: Find the plane z = a + bx + cy that best fits the data points (0, -4, 0), (5, 0, 0), (4, –1, 1), (1,…

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Q: Find the projection of d on the plane P if a 4 and P has the equation x + y+ z = 1.

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Q: Find a plane containing the point (7,-8,8) and the line F(t) = +t

A: Given that, the point is 7,-8,8 and the line is r→t=-6,-8,-7+t6,8,4. From the given line, the point…

Q: Find the intersection of the plane: x - z = 6, and the line r(t) = (1,0, -,1) + t(4,9,2)

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Q: Example 8: Find the point where the line 2t, and z = -3t intersects the plane x = 1 + t, y = –…

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Q: Find the distance between the plane P:8x+3y+9%= 0 and the point (1,-5,20)

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Q: Find the general equation of the plane π  that passes through the origin and is perpendicular to the…

A: Suppose a plane passing through the points x1,y1,z1 and a vector perpendicular to the plane is…

Q: Find a plane containing the point (6, -4,-3) and the line (t) = +t 270

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Q: 1 2x 3y + z = 0 and 2 +t - 1 1

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Q: Find the point of intersection, if any, of the plane T: 4x + 2y – 2z=1 and the line l : 3 +t 1

A: Point of intersection of line and plane

Q: Find the plane z = a + bx + cy that best fits the data points (0, -4, 0), (5, 0, 0), (4, -1, 1), (1,…

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Q: Find the equation of the plane containing the point P(1, 1, –1) and the line -4+t 1- t 1+t

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Q: 4) Find the equation of the plane that passes through the point (6,0, 2) and contains the line x = 4…

A: The plane that passes through the point (6, 0, 2) and contains the line x = 4 − 2t, y = 3 + 5t, z =…

Q: Sketch the following planes in the window [0, 5] × [0, 5] × [0, 5]. The plane parallel to the…

A: First obtain the plane that parallel to the xz-plane containing the point (1, 2, 3).…

Q: If 1x, y2 are the coordinates of a point P in the xy-plane, then x is called the _________  of P and…

A: If (x,y) are the coordinates of a point P in the xy-plane, then x is called the _________  of P and…

Q: 10) Find the point of intersection of the lines L4:x = 1+ 2t, y = 2 + 3t, z = 3 + 4t and L2: x = 2 +…

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Q: Determine the general equation of the plane N passing through the point P(3, –1,2) and parallel to…

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Q: Find the equation of the plane with the given description. Passes through P = (3, 5, −9) and is…

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Q: Find an equation of the plane containing the point (1,2,-2), and the line: x= 2t, y= 3-t, and z…

A: To find the equation of plane we need to find its normal vector. To find the normal vector, we…

Q: Find the equation of the plane: The plane that passes through the point (3, 4, 5) and contains the…

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Q: Find a plane containing the point (-3,3,-3) and the line F(t) = +t

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Q: Let L be the line that passes through the point  (1,3,1 ) and orthogonal to the plane 3x-2y + 5z =…

A: In the question we have to find interacting points at which line and plane intersect.

Q: Determine the equation of the plane through the points (1, 0, 3) and the line l : ¯r = (3, 4, 1) +…

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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 equation of the plane that contains the following two lines: I = -1+t, L1 : y = -t, I = -1,…

A: To find the equation of plane containing both the lines

Q: Find the point of intersection of the lines x = t, y = -t + 2, z = t + 1, and x = 2s + 2, y = s + 3,…

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Q: 4. Find an equation for the plane in R³ spanned by the points (1,5, 1) and (1, 4, 2).

A: The two vectors are given as v1 = 151v2 = 142 Let xyz∈span{v1 , v2} Then xyz = mv1+nv2= m151+n142=…

Q: Find an equation for the plane that passes through the point (1, 3, -5) and is perpendicular to the…

A: Given point is (1,3,-5). The scalar equation of a plane that passes through point (a, b, c) and has…

Q: Find an equation of the plane. The plane that passes through the point (3, 4, 5) and contains the…

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Q: Find the point at which the line (-2, 2, 1) + t(5, –4, 3) intersects the plane -4x – 4y + z = -5.

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Q: Find the equation of the plane which contains the line ( 2 + t , 3 − t , 4 + 2 t )and the point ( 0…

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Q: Find and describe the intersection of the following line and plane: T:2x+3y-4z =-7 and r = (4,2,5)…

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Q: Find the equation of the plane through the point P = (3, 2, 5) and parallel to the plane a + 3y+ 3z…

A: Given: Line is parallel to x+3y+3z = 5 Equation of the plane parallel to the plane x+3y+3z = 5 is…

Q: 2. Find an equation of the plane containing the point P (0, -) and the line determined by the two…

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Q: 2) Find the equation of the plane that is orthogonal to the planes 3x-2y-2z=1 and x-3y-10z = 4 and…

A: Given: The equation of two planes 3x-2y-2z=1 and x-3y-10z=4 and the equation of one line…

Q: Find the equation of the plane whose has the line x=2-3t, y=3 + t, z = 5t and passes through the…

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Q: Find the projection of a on the plane P if a 4 and P has the equation x + y + z = 1.

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Q: The line L(t) = (2+1,8-31, 2t-6) intersects the plane 2y - 3x + 3z = -11 at the point

A: Given, the line             Lt=2+t, 8-3t, 2t-6                                   ...(1) Intersects…

Q: Determine the point of intersection, if such a point exists, for the line €: (x, y, z) = (3,– 1, 7)…

A: Given: Parametric equation of line l: (x,y,z)=(3,-1,7)+r(-5,2,1) Equation of plane: 2x+3y+4z=24

Q: 1. Find the plane which contains the point M (4, 5,0) and the line 2+3 = ¹ = 2²

A: To find the plane which contains the point M4, 5, 0 and the line x+35=y-4-3=z-22.

Q: Which of the following is an equation for the line through the point (2, 3) and perpendiçular to the…

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Q: An equation of a plane P passes through the point (0, 0, 3) and perpendicular to the line L:x =1 –…

A: Remember that the vector (l, m, n) is parallel to the line with parametric equationx=a+lt, y=b+mt,…

Q: a) Find the equation of the plane containing the points P(3, 1, 5), Q(- 1,– 1,– 1), and R(– 2,2, 4).

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Q: Find the equation of the plane through the point P = (5, 3, 3) and parallel to the plane 3y – 5x –…

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Q: 6.) Consider the points P(-1,2,5), Q(2, –3, 0) and (i) Write an equation of the plane containing t

A: Equation of plane: We know that, the equation of plane that passes through point (x, y, z) is given…