The helix  r(t) =   cos(πt/2), sin(πt/2), t    intersects the sphere  x2 + y2 + z2 = 2  in two points. Find the angle of intersection at each point.

Q: Find the point where the helix r(t) = (cos(t), sin(t), t) intersects the plane z = (х, у, 2) %3D

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Q: Find, correct to four decimal places, the length of the curve of intersection of the cylinder 9x2 +…

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Q: Find the points on the surface y2 36 + xz that are closest to the origin. (х, у, 2) (smaller…

A:

Q: Consider the unit sphere x² + y² + z² = 1. Find the average distance from its north pole to a point…

A: Given: x2+y2+z2=1 It is equation of sphere of radius r=1 unit. We have to find the average distance…

Q: If a parametric surface given by r₁ (u, v) = f(u, v)i + g(u, v)j + h(u, v)k and -2 ≤ u ≤ 2, -3 ≤ v ≤…

A:

Q: 6) Find the equation of the osculating plane of the helix x = sin 3t, y = 2t, z = cos 3t at the…

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Q: The distance between the centers-of the following two spheres: x²-22x+y2-58y+z?=-961,…

A:

Q: Find the angle between the two planes 6x − 3y + z = 5,  −x + y + 5z = 5 and find a set of parametric…

A: We have given the equation of planes; 6x − 3y + z = 5 −x + y + 5z = 5 Hence the normal vectors of…

Q: Find the standard equation of the sphere with center (−3, 2, 4) that is tangent to the plane given…

A: Radius (r) will be the distance from the center to the tangent plane. That is the distance between…

Q: find the areas of surfaces generated by revolving the curves about the given axes

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Q: Integrate counterclockwise dz, C the circle with z2 + 12z + 35 center –5 and radius 2. dz z2 + 12z +…

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Q: An angle with its vertex at the origin of an xy-coordinate plane and with initial side on the…

A: We know, An angle with its vertex at the origin of an x-y-coordinate plane and with initial side on…

Q: Find a formula for the distance from the point P(x, y, z) to the yz-plane.

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Q: The axis of symmetry of the cylinder is parallel to the z-axis. х-ахis. O y-axis.

A:

Q: Find the surface area of the portion of the sphere generated by revolving the curve y = 1- x², 0sis…

A: y=1-x2,  0≤x≤12 Differentiating with respect to x, we get dydx=121-x2·0-2x=-x1-x2

Q: and the plane - 3 X + Z Intersect along a line. Find Parametnic desenetion of this la

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Q: Use cos(t) and sin(t), with positive coefficients, to parametrize the intersection of the surfaces…

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Q: Find the center C and the radius a for the spheres 2x2 + 2y2 + 2z2 + x + y + z = 9

A: General equation of a sphere, x2+y2+z2+2ux+2vy+2wz+d=0 Centre of sphere = -u, -v, -w Radius of…

Q: Use cos(t) and sin(t), with positive coefficients, to parametrize the intersection of the surfaces…

A:

Q: Find the equation of the sphere of radius 8 centered at (1, 3, 6). O x² + y² +z² = 46 O x² + y² + z²…

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Q: Find the angle between the two planes 3x + 2y − z = 7,  x − 4y + 2z = 0 and find a set of parametric…

A: Known fact: The angle between two planes a1x+b1y+c1z+d1=0  and a2x+b2y+c2z+d2=0 is given by…

Q: · What is the shortest distance from the surface xy + 6x + z = 41 to the origin? distance =

A: The surface is given as , xy+6x+z2=41 find the shortest distance from the surface to the origin.

Q: Find parametric equations for the circle x2 + y2 = a2, using as parameter the arc length s measured…

A: The given equation of circle is,

Q: PICTURE

A: We are given that the cone ax2+by2+z2 = 0 and the ellipsoid 2x2+4y2+z2 = 42 intersect and are…

Q: What is the definition of a parabolic cylinder?

A: The parabolic cylinder functions are the solutions to the differential equation d2ydx2 + ax2 + bx +…

Q: Find the equation of spheres passing through the circle x+y+z'-6x-2z+5 =0, y = 0 and touching the…

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Q: Prove that the product of the perpendicular distances F1R1 and F2R2 from the foci to a tangent line…

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Q: . Find a parametrization for the circle (x – 2)² + y² = 1 starting at (1, 0) and moving clockwise…

A: Consider the equation of circle

Q: Which type of surface does the curvex= cos(3t),y= sin(3t),z=t, lie on?

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Q: find an equation for the plane that is tangent tothe given surface at the given point. z = 4x2 + y2,…

A: It is given that, z = 4x2 + y2, at the point (1, 1, 5) We have to find the equation of tangent…

Q: Find the equation of the tangent plane to the sphere x² + y2 + z² = 1 at the point (1/ V3, 1/V3, 1/…

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Q: If a parametric surface given by r1(u, v) = f(u, v)i + g(u, v)j+ h(u,v)k and –1< u<1, –2 < v< 2, has…

A: We will find out the required value.

Q: Find the area of the surface when y = vx, from (4, 2) to (9, 3) is rotated about the x-axis.

A: Surface area formula for rotating about x-axis is S=∫2πydS where dS=1+dydx2dx

Q: The bearing of x from y is 045 and the bearing of z from y is 145 ,where x, y and z are three point…

A: Rules for solving Bearing and Distances: The reading in bearing is taken from North as per the…

Q: (a) Obtain the equation of the tangent plane and the condition for tangency to the conicoid Ar? +…

A: Conicoid given is- Ax2+By2+cz2=1 at the point (α,β,γ) We have to obtain the equation of the tangent…

Q: Find the standard equation of the sphere with center (-6, 2, 7) that is tangent to the plane given…

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Q: Let C be the curve of intersection of the parabolic cylinder x = 2y, and the surface 3z = xy. Find…

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Q: The circle of radius 1 centered at (-3, 4,1) and lying in a plane parallel to the yz-axis is a)…

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Q: ston Compute the length of the circle of radius r defined parametrically by x = 3 cost, y = 3 sint ;…

A: given that x=3 cost , y=3 sint; 0≤t≤2π find out the length of the circle of radius r i.e. arc length…

Q: Find the length of the circle of radius r defined parametrically by x=rcost, y=rsint,0≤t≤2π.

A: Here I am attaching image so that you understand each and every step.

Q: Find the angle of tilt θ with the horizontal so that the contact force at B will be equal to 0.76…

A: Draw the free body diagram of the block.

Q: Find the points on the cone z = x° + y° that are closest to the point (1,-1,0) %3D +1

A: Given,

Q: Compute the surface area of the surface generated by revolving one arch of the cycloid…

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Q: The tangent plane to the elliptic paraboloid z = -2x - 7y? at the point (-3, -4) is

A: Consider the provided question, The equation of the tangent plane to the elliptic paraboloid at the…

Q: Find a formula for the distance from the point P(x, y, z) to the xz-plane

A: The distance between two points x1,y1z1 and x2,y2z2: d=x2-x12+y2-y12+z2-z12 The distance from the…

Q: Give an equation of the surface of revolution generated by revolving x In z about the z-axis.

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Q: A flat, very thin table has a circular hole with radius 1 and a circular hole with radius 2. The…

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Q: Find the equation of the line tangent to the intersection of the surface z = arctan ry with the…

A: The tangent line to a plane curve at a particular position is the straight line that "just touches"…

Q: Find the point lying on the intersection of the plane x + + z = 0 and the sphere x2 + y2 + z2 = 25…

A: According to the given information, it is required to find a point lying on the intersection of the…

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.