Explanation Given data: The certain plane is given as, 2 x + 3 y + 4 z = 16 (1) Calculation: Let the three points on the plane are P , R and Q . Substitute 0 for x and 0 for y in equation (1). 2 ( 0 ) + 3 ( 0 ) + 4 z = 16 4 z = 16 z = 4 So, the point is written as, P = ( 0 , 0 , 4 ) Here, P is the point on the given plane. Substitute 0 for z and 0 for y in equation (1). 2 x + 3 ( 0 ) + 4 ( 0 ) = 16 2 x = 16 x = 8 So, the point is written as, R = ( 8 , 0 , 0 ) Here, R is the point on the given plane. Substitute 0 for x and 0 for z in equation (1). 2 ( 0 ) + 3 y + 4 ( 0 ) = 16 3 y = 16 y = 16 3 So, the point is written as, Q = ( 0 , 16 3 , 0 ) Here, Q is the point on the given plane. The vector A from point ( 0 , 0 , 4 ) to ( 8 , 0 , 0 ) is written as, A = x ^ ( 8 − 0 ) + y ^ ( 0 − 0 ) + z ^ ( 0 − 4 ) A = x ^ 8 − z ^ 4 (2) The vector B from point ( 0 , 0 , 4 ) to ( 0 , 16 3 , 0 ) is written as, B = x ^ ( 0 − 0 ) + y ^ ( 16 3 − 0 ) + z ^ ( 0 − 4 ) B = y ^ 16 3 − z ^ 4 (3) The cross product of vector A and B is written as, A × B = | x ^ y ^ z ^ 8 0 − 4 0 16 3 − 4 | = x ^ ( 0 + 64 3 ) − y ^ ( − 32 − 0 ) + z ^ ( 128 3 − 0 ) = x ^ ( 64 3 ) + y ^ ( 32 ) + z ^ ( 128 3 ) Let the orthogonal vector to the plane will be vector C so, the cross product of vector A and B will be the vector C and it is written as, C = A × B Substitute x ^ ( 64 3 ) + y ^ ( 32 ) + z ^ ( 128 3 ) for A × B in the above expression

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Author: ULABY

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Spherical Coordinate System

The spherical coordinate system is a three-dimensional coordinate system for space where the position of a point is specified by three numbers- the radial distance of that point from a fixed source, its polar angle measured from a certain zenith, and the…

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Chapter 3, Problem 15P

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The unit vector normal to the surface in the direction away from the origin.

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EBK FUNDAMENTALS OF APPLIED ELECTROMAGN

Ch. 3.1 - When are two vectors equal and when are they...Ch. 3.1 - Prob. 2CQ Ch. 3.1 - If AB = 0, what is AB?Ch. 3.1 - If AB = 0, what is AB?Ch. 3.1 - Is A(BC) a vector triple product?Ch. 3.1 - If AB = AC, does it follow that B = C?Ch. 3.1 - Find the distance vector between P1 = (1, 2, 3)...Ch. 3.1 - Find the angle AB between vectors A and B of...Ch. 3.1 - Prob. 3E Ch. 3.1 - Vectors A and B lie in the y-z plane and both have...

Ch. 3.1 - If AB=AC, does it follow that B = C?Ch. 3.2 - A circular cylinder of radius r = 5 cm is...Ch. 3.3 - Why do we use more than one coordinate system?Ch. 3.3 - Prob. 8CQ Ch. 3.3 - Prob. 9CQ Ch. 3.3 - How is the position vector of a point in...Ch. 3.3 - Prob. 7E Ch. 3.3 - Prob. 8E Ch. 3.4 - Prob. 9E Ch. 3.4 - Find the directional derivative of V=rz2cos2 along...Ch. 3.4 - Prob. 11E Ch. 3.4 - Prob. 12E Ch. 3.5 - Given A=e2y(xsin2x+ycos2x), find A.Ch. 3.5 - Given A=rrcos+rsin+z3z , find A at (2,0, 3).Ch. 3.5 - If E=RAR in spherical coordinates, calculate the...Ch. 3.5 - Verify the divergence theorem by calculating the...Ch. 3.5 - Prob. 17E Ch. 3.6 - Find A at (2, 0, 3) in cylindrical coordinates for...Ch. 3.6 - Find A at (3, /6, 0) in spherical coordinates for...Ch. 3.7 - What do the magnitude and direction of the...Ch. 3.7 - Prob. 12CQ Ch. 3.7 - Prob. 13CQ Ch. 3.7 - Prob. 14CQ Ch. 3.7 - What is the meaning of the transformation provided...Ch. 3.7 - Prob. 16CQ Ch. 3.7 - Prob. 17CQ Ch. 3.7 - When is a vector field conservative?Ch. 3 - Prob. 1P Ch. 3 - Given vectors A=x2y3+z, B=x2y+z3, and C=x4+y2+z2,...Ch. 3 - Prob. 3P Ch. 3 - Prob. 4P Ch. 3 - Given vectors A=x+y2z3, B=x2y4, and C=y2z4, find...Ch. 3 - Given vectors A=x2y+z3 and B=x3z2, find a vector C...Ch. 3 - Given A=x(x+2y)y(y+3z)+z(3xy), determine a unit...Ch. 3 - By expansion in Cartesian coordinates, prove: (a)...Ch. 3 - Find an expression for the unit vector directed...Ch. 3 - Prob. 10P Ch. 3 - Prob. 11P Ch. 3 - Prob. 12P Ch. 3 - A given line is described by x+2y=4. Vector A...Ch. 3 - Prob. 14P Ch. 3 - Prob. 15P Ch. 3 - Given B=x(z3y)+y(2x3z)z(x+y), find a unit vector...Ch. 3 - Find a vector G whose magnitude is 4 and whose...Ch. 3 - A given line is described by the equation: y=x1....Ch. 3 - Vector field E is given by E=R5Rcos12Rsincos+3sin....Ch. 3 - Prob. 20P Ch. 3 - Prob. 21P Ch. 3 - Prob. 22P Ch. 3 - Prob. 23P Ch. 3 - Prob. 24P Ch. 3 - Use the appropriate expression for the...Ch. 3 - Prob. 26P Ch. 3 - A section of a sphere is described by 0 R 2, 0 ...Ch. 3 - A vector field is given in cylindrical coordinates...Ch. 3 - At a given point in space, vectors A and B are...Ch. 3 - Given vectors...Ch. 3 - Prob. 31P Ch. 3 - Prob. 32P Ch. 3 - Transform the vector A=Rsin2cos+cos2sin into...Ch. 3 - Transform the following vectors into cylindrical...Ch. 3 - Transform the following vectors into spherical...Ch. 3 - Find the gradient of the following scalar...Ch. 3 - For each of the following scalar fields, obtain an...Ch. 3 - The gradient of a scalar function T is given by...Ch. 3 - Prob. 39P Ch. 3 - For the scalar function V = xy2 z2, determine its...Ch. 3 - Evaluate the line integral of E=xxyy along the...Ch. 3 - Prob. 42P Ch. 3 - Prob. 43P Ch. 3 - Each of the following vector fields is displayed...Ch. 3 - Prob. 45P Ch. 3 - For the vector field E=xxzyyz2zxy, verify the...Ch. 3 - For the vector field E=r10erz3z, verify the...Ch. 3 - A vector field D=rr3 exists in the region between...Ch. 3 - For the vector field D=R3R2, evaluate both sides...Ch. 3 - For the vector field E=xxyy(x2+2y2), calculate (a)...Ch. 3 - Repeat Problem 3.50 for the contour shown in Fig....Ch. 3 - Verify Stokess theorem for the vector field...Ch. 3 - Prob. 53P Ch. 3 - Prob. 54P Ch. 3 - Verify Stokess theorem for the vector field B = (r...Ch. 3 - Prob. 56P Ch. 3 - Prob. 57P Ch. 3 - Find the Laplacian of the following scalar...

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