﻿MTH282 Two vectors are said to be equal if their --------- are equal. magnitudes and directions Two vectors A, B, are said to collinear ( linearly dependent) if there are scalar a and b such aA + bB = 0 implies ----- a and b are not all zero Collinear vectors are said be -------- linearly dependent Find the unit vector in the direction of the vector 3i + 2j + 6k. (3i + 2j + 6k)/7 The following are true about scalar product of vectors accept ----- j. j = 1 Given that v = 2i + j + 5k and u = 3i -4j + k, find v.u 7 Find the direction cosines of the vector 3i + 4j 3/5, 4/5 Find sum of direction cosines of the vector 3i + 2j + 6k. 1/7 Given that vector D= 2i - 3j + 5k and P = 4i + j + 6k find D.P 35 If vector OC = 2i + j and vector OB= 6i -2j, find the magnitude of vector CB 5 Find the vector product of A = i + j+K and B = 3i +3j+3K 0 Find the acute angle (to the nearest degree) angle between the vectors a=-3i+4j and b=5i+12j. 59.49° Find a vector that is of magnitude 6 units and is collinear to the vector i+j. 32i+32j Given that A =2i + j-3k, B = i- 2j+k C= i +j -4k. Evaluate A. (BC). 10 Find the unit vector perpendicular to the plane of the vectors A=3i-2j+4k and B= i+j-2k 15(2i+j) An object moves in a straight curve R t32t) i -3e2t j +2sin 5tk, Find its velocity at time t=1. 5i +0.812 j+ 2.837k If the scalar product of vectors 2i - j -4k and 5i - 2j –mk is four, find m. -2 Find the angle between the vectors 2i-4 j +k and 2i -3j -3k. cos-1(0.6048) Determine s such that the vectors s i +4 j +4k and –2 i + 5j +sk perpendicular. -10 Find the value of a that makes the vectors 5 i -a j +2k and i + 3j +5k perpendicular. 5 Determine n such that the vectors 5i +n j -5k and -4 i + nj +k are perpendicular. 5 Determine the unit vector parallel to the sum of vectors 3i +4 j -5k and i + 8j +9 12i+4j+3k13 Find the scalar product of the vector -2i-j-2k and the unit vector parallel to the vector 4i+3j-12k 1 Find the vector product of v = (1, 0,2) and u = (2, 4, 3). (-8,1, 4) Find the gradient of the scalar field ∅ x, y, z=x3yz2 at the point (1,1,1) 3i + j +2k If that ∅=x4y+z2, find the Laplacian of ∅ 2(6x2 +1) Given that R=xzi-y2j+2x2yk , find ∇.(∇xR) 0 Given that R=xzi-y2j+2x2yk , find the divergence of R z-2y Given that ∅=3y2x+x2z2, find ∇∅ (3y2+2xz2)i+6yxj+2zx2k If ∅=3y2x+x2z2 , find ∇2∅. 2z2+6x+2x2 Given that =yxi+yx2j+2zyk , find (∇xA) 2zi+x(2y-1)k Given that T=y2x2i+x2z2 j+y2z2k. find curl of T. 2z (yz-x2)i+2x(z2-yx)k Given that T=y2x2i+x2z2 j+y2z2k. find curl of T at point (1, -1,1) -4i+4k The scalar product of vectors a and b, where θ is the angle between them, is ......... |a||b| cos θ If that ∅=x2z-yz2, find the Laplacian of ∅ 2z-2y If that ∅=x2y2z2, find the Laplacian of ∅. 2(y2z2+x2z2+x2y2) Given thatA =yxi+yx2j+2zyk , find ∇x(∇xA) 2xi-(2y-3)j Let =yxi+xzj+zyk , find ∇x(∇xE) . -2j Find the Jacobian of the transformation x=Rsin⁡θ⁡cos⁡∅, y=Rsin⁡θ sin⁡∅ and z=R⁡cos⁡∅ with respect to R , θ and ∅ -R2sin⁡θ Determine the Jacobian of x=u+v, y=2v with respect to u and v 2 Given that u =x+2y and v=x - y, find the Jacobian of the transformation x and y with respect to u, and v -1/3 Find the Jacobian of x=u+v +s, y=2v-s and z=u-v with respect to u, v and s -5 Given that x=2s – w, y=v+s+2w and z = 2v-s -w find the Jacobian of the transformation x and y with respect to v , s and w5 13 Let ∂x, y∂u, v be the Jacobian of the transformation x and y with respect to u and v, then ∂x, y∂u, v is equal to -------- -∂y, x∂u, v Expand 1-i1+i4 1 Let F(z)= 7-z1-z2 and z =1+2i. Find F(z). 1+i/2 Given that R =y2zx2i+xj-y4zx3k , find ∇.R . 2xy2zi+y4x3k Given that u =x+4y and v=x-3y, find the Jacobian of the transformation x and y with respectt to u , and v -2 Given that u =x+2y and v=x - y, find the Jacobian of the transformation x and y with respect to u and v -1/7 Given that =zxi+zyj+yxk , find ∇x(∇xA) 2k Two vectors are said to be equal if their --------- are equal. magnitudes and directions Two vectors A, B, are said to collinear ( linearly dependent) if there are scalar a and b such aA + bB = 0 implies ----- a and b are not all zero Collinear vectors are said be -------- linearly dependent Find the unit vector in the direction of the vector 3i + 2j + 6k. (3i + 2j + 6k)/7 The following are true about scalar product of vectors accept ----- j. j = 1 Given that v = 2i + j + 5k and u = 3i -4j + k, find v.u 7 Find the direction cosines of the vector 3i + 4j 3/5, 4/5 Find sum of direction cosines of the vector 3i + 2j + 6k. 1/7 Given that vector D= 2i - 3j + 5k and P = 4i + j + 6k find D.P 35 If vector OC = 2i + j and vector OB= 6i -2j, find the magnitude of vector CB 5 Find the vector product of A = i + j+K and B = 3i +3j+3K 0 Find the acute angle (to the nearest degree) angle between the vectors a=-3i+4j and b=5i+12j. 59.49° Find a vector that is of magnitude 6 units and is collinear to the vector i+j. 32i+32j Given that A =2i + j-3k, B = i- 2j+k, C= i +j -4k. Evaluate A. (BxC). 10 Find the unit vector perpendicular to the plane of the vectors A=3i-2j+4k and B= i+j-2k 15(2i+j) 5i +0.812 j+ 2.837k If the scalar product of vectors 2i - j -4k and 5i - 2j –mk is four, find m. -2 Find the angle between the vectors 2i-4 j +k and 2i -3j -3k. cos-1(0.6048) Determine s such that the vectors s i +4 j +4k and –2 i + 5j +sk perpendicular. -10 Find the value of a that makes the vectors 5 i -a j +2k and i + 3j +5k perpendicular. 5 Determine n such that the vectors 5i +n j -5k and -4 i + nj +k are perpendicular. 5 Determine the unit vector parallel to the sum of vectors 3i +4 j -5k and i + 8j +9 12i+4j+3k13 Find the scalar product of the vector -2i-j-2k and the unit vector parallel to the vector 4i+3j-12k 1 Find the vector product of v = (1, 0,2) and u = (2, 4, 3). (-8,1, 4) Find the gradient of the scalar field ∅ x, y, z=x3yz2 at the point (1,1,1) 3i + j +2k If that ∅=x4y+z2, find the Laplacian of ∅ 2(6x2 +1) Given that R=xzi-y2j+2x2yk , find ∇.(∇xR) 0 Given that R=xzi-y2j+2x2yk , find the divergence of R z-2y Given that ∅=3y2x+x2z2, find ∇∅ (3y2+2xz2)i+6yxj+2zx2k If ∅=3y2x+x2z2 , find ∇2∅. 2z2+6x+2x2 Given that =yxi+yx2j+2zyk , find (∇xA) 2zi+x(2y-1)k Given that T=y2x2i+x2z2 j+y2z2k. find curl of T. 2z (yz-x2)i+2x(z2-yx)k Given that T=y2x2i+x2z2 j+y2z2k. find curl of T at point (1, -1,1) -4i+4k The scalar product of vectors a and b, where θ is the angle between them, is ......... |a||b| cos θ If that ∅=x2z-yz2, find the Laplacian of ∅ 2z-2y If that ∅=x2y2z2, find the Laplacian of ∅. 2(y2z2+x2z2+x2y2) Given that A =yxi+yx2j+2zyk , find ∇x(∇xA) 2xi-(2y-3)j Let E=yxi+xzj+zyk , find ∇x(∇xE) . -2j Find the Jacobian of the transformation x=Rsin⁡θ⁡cos⁡∅, y=Rsin⁡θ sin⁡∅ and z=R⁡cos⁡∅ with respect to R , θ and ∅ -R2sin⁡θ Determine the Jacobian of x=u+v, y=2v with respect to u and v 2 Given that u =x+2y and v=x - y, find the Jacobian of the transformation x and y with respect to u, and v -1/3 Find the Jacobian of x=u+v +s, y=2v-s and z=u-v with respect to u, v and s -5 Given that x=2s – w, y=v+s+2w and z = 2v-s -w find the Jacobian of the transformation x and y with respect to v , s and w5 13 Let ∂x, y∂u, v be the Jacobian of the transformation x and y with respect to u and v, then ∂x, y∂u, v is equal to -------- -∂y, x∂u, v Expand 1-i1+i4 1 Let F(z)= 7-z1-z2 and z =1+2i. Find F(z). 1+i/2 Given that R =y2zx2i+xj-y4zx3k , find ∇.R . 2xy2zi+y4x3k Given that u =x+4y and v=x-3y, find the Jacobian of the transformation x and y with respect to u , and v -2 Given that u =x+2y and v=x - y, find the Jacobian of the transformation x and y with respect to u and v -1/7 Given thatA =zxi+zyj+yxk , find ∇x(∇xA) 2k A vector V with a unit vector a and magnitude k is written as V = ----- *Ka* The unit vector in the direction of the resultant of vectors 2i -j + k and i + j + 2k is...... *i + k* If the scalar product of vectors i - j -k and 3i -2j –ak is eight, find a. *3* If A.B =0 then the angle between vectors is -------- *90* If AxB =0 and A and B are not null vectors, then A and B are ------- *Parallel* Find the acute angle between the vectors a=-4i-3j and b=5i-12 to nearest degree. *58* The vector product of a=2i+j+k and b=i+3j-2k is ----- *(17)-i+5j+5k* Determine the acute angle between the vectors 2i+4j and 5i-4j to the nearest degree. *78°* Let ∅ (x, y, z) = constant c be an equation of a surface then, ∇∅ is ------to this surface. *Normal* If ∅ x, y, z=x4y6z4+xy . determine ∇x(∇∅) *Zero* Let ∅ x, y, z=x3y2z4 . determine curl of ∇∅ *0* Find the gradient of the scalar field ∅ x, y, z=x3y2z4 at the point (1,1,1) *3i+2j+4k* Let ∂x, y∂u, v be the Jacobian of the transformation x and y with respect to u and v. Then ∂x, y∂u, u is equal to -------- *0* Let ∂x, y∂u, v be the Jacobian of the transformation x and y with respect to u and v. Then ∂x, k∂u, v (where k is constant) is equal to -------- *0* Determine the vector product of the vectors u=3i-j+k and v=4i+2j-k. *-i+7j+10k* Determine the scalar product of the vectors A =4i +2 j- 6k and B = i+ 6j+k *10* Given that A =2i + j-3k, B = i- j+2k, C= i +3j -k. Evaluate A. (BxC ) *-19* Find the vector perpendicular to the plane of the vectors A= i-j+k and B= i+j-2k *-i+3j+2k* Given that A =2i + j-3k, B = i- j+2k C= i +3j -k. Evaluate C. (AxB) *-19* An object moves in a straight curve G=(t3+2t2+2t)i+3e(t-1)j+2tk. Find its acceleration at time t=1. *10i +3j* Find the unit vector in the direction of the sum of the vectors i +6j and 2i - 2j . *3i/5+4i/5* An object moves in a straight curve R=(3t2+t)i+3e(t-2)j+2k Find its velocity at time t=2. *13i +3j* Find the gradient of the scalar field ∅ x, y, z=4xzy2 at the point (1, 1, 2) *8i +16j +4k* Given that E=xzi-y2j+2x2yk , find ∇.E *4xy* Given that R=x+zi-y2j+2x2yk , find ∇.∇xR *0* Given that T=y2i+z2 j+x2k. find curl of T. *2zi+2xj-2ky* Given that P=y2i+z2 j+x2k.. find curl of P at point (2, 2, 2) *4i+4j+4k* Let the scalar product of vectors xi +3 j -5k and xi + j -2k be thirty eight. Find x. *5* The scalar product of vectors 2i + cj +6k and 3i + 5j -6k.is five, find c. *7* If that ∅=x2y2z2, find the Laplacian of ∅ at (1, -1, -1). *6* Given that A =yxi+yx2j+2zyk , find curl curl A at (3, -1, 5) *6i+5j* Let E=yxi+xzj+zyk , find magnitude of curl E *2* Find the Jacobian of the transformation x=(r+2)sin⁡θ , y=(r+2)cos⁡θ with respect to r and θ * -(r+2)* Determine the Jacobian of x=vcos⁡θ , y=vsin⁡θ with respect to v and θ *V* Determine the Jacobian of x=3u+2v, y=v with respect to u and v *3* Find the Jacobian of x=u-v +s, y=2u+v-s and z=v+s with respect to u, v and s *6* If A¯ is the conjugate of the complex number A, determine A¯ +A 2Re(z)* Evaluate 1-i1+in where n is a positive even positive integer. *1* Evaluate 1-i1+in where n is a positive odd positive integer. *-i* Given that F(z)= 2-z1-z and z =1+i, find F(z) *-1+2i* Given that F(z)= 4+z1-z and z =1-i, find F(z) *-1+5i* Expand (1+2i)(1+i)(1-i). *2+4i* Find the real part of the complex number 4+i4-i2 *15/17* If z=2(cos⁡π6+isin⁡π6), find z6 *-64* Evaluate 1+i1-i8 *1* Evaluate 1-i1+in where n is a positive odd positive integer. *-i* Given that u =3x+y and v=x -2y, find the Jacobian of the transformation x and y with respect to u, and v *1* Evaluate 1-i1+in where n is a positive even positive integer. *1* Given that u =x+y and v=2x-y, find the Jacobian of the transformation x and y with respectt to u , and v *-1/3* Evaluate 1-i1+in where n is a positive even positive integer. *1*