Hmm.Doc
Pandit Deendayal Petroleum University
School of Technology
Department of Sciences Tutorial 1 Vector Operations
1.
If A = 2i - 3j + 4k and B = i + 4j – 2k, find (a) A x B (b) B x A (c) (A + B) x (A – B). If A = 3i - j + 2k and B = 2I + j – k and C= I -2j + 2k, find (a) (A x B) x C (b) A X (B X C). Determine the unit vector perpendicular to the plane of A = 2i -6j - 3k and B = 4I +3j – k. Find the directional derivative of Φ = x2yz +4xz2 at (1,-2,-1) in the direction 2i – j - 2k. (a) In what direction from the point (2,1,-1) is the directional derivative of Φ = x2yz3 a maximum? (b) What is the magnitude of maximum? Find the angle between the surfaces x2 +y2 +z2 = 9 and z = x2 + y2 – 3 at the point (2,-1, 2). If A = x2z i – 2y3z2 j + xy2z k, find ∇.A at the point (1,-1,1). Given Φ = 2x3y2z4. Find (a) ∇.∇ Φ Prove that ∇2 (1 / r) = 0. Prove that (a) ∇. (A + B) = ∇. A + ∇. B Prove that ∇. (r / r3) = 0. If A = xz3i - 2x2yzj + 2yz4k, find ∇ x A at the point (1,-1,1). If A = x2yi - 2xzj + 2yzk, find curl curl A. Prove that (a) ∇ x (A + B) = ∇ x A + ∇ x B (b) ∇ x ΦA = (∇Φ) x A + Φ (∇ x A) Evaluate∇. (A x r), if ∇ x A = 0 Prove (a) ∇ x (∇Φ) = 0 (b) ∇. (∇ x A) = 0. If A = 2yz i - x2y j + xz2 k, B = x2 i + yz j -xy k and Φ = 2x2yz3, find (a) (A.∇) Φ (b) A.∇Φ (c) (B.∇)A (d) (A x ∇) Φ (e) A x ∇Φ (b) ∇.(ΦA) = (∇Φ).A + Φ (∇.A) (b) Show that ∇.∇ Φ = ∇2Φ
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School of Technology
Department of Sciences Tutorial 1 Vector Operations
1.
If A = 2i - 3j + 4k and B = i + 4j – 2k, find (a) A x B (b) B x A (c) (A + B) x (A – B). If A = 3i - j + 2k and B = 2I + j – k and C= I -2j + 2k, find (a) (A x B) x C (b) A X (B X C). Determine the unit vector perpendicular to the plane of A = 2i -6j - 3k and B = 4I +3j – k. Find the directional derivative of Φ = x2yz +4xz2 at (1,-2,-1) in the direction 2i – j - 2k. (a) In what direction from the point (2,1,-1) is the directional derivative of Φ = x2yz3 a maximum? (b) What is the magnitude of maximum? Find the angle between the surfaces x2 +y2 +z2 = 9 and z = x2 + y2 – 3 at the point (2,-1, 2). If A = x2z i – 2y3z2 j + xy2z k, find ∇.A at the point (1,-1,1). Given Φ = 2x3y2z4. Find (a) ∇.∇ Φ Prove that ∇2 (1 / r) = 0. Prove that (a) ∇. (A + B) = ∇. A + ∇. B Prove that ∇. (r / r3) = 0. If A = xz3i - 2x2yzj + 2yz4k, find ∇ x A at the point (1,-1,1). If A = x2yi - 2xzj + 2yzk, find curl curl A. Prove that (a) ∇ x (A + B) = ∇ x A + ∇ x B (b) ∇ x ΦA = (∇Φ) x A + Φ (∇ x A) Evaluate∇. (A x r), if ∇ x A = 0 Prove (a) ∇ x (∇Φ) = 0 (b) ∇. (∇ x A) = 0. If A = 2yz i - x2y j + xz2 k, B = x2 i + yz j -xy k and Φ = 2x2yz3, find (a) (A.∇) Φ (b) A.∇Φ (c) (B.∇)A (d) (A x ∇) Φ (e) A x ∇Φ (b) ∇.(ΦA) = (∇Φ).A + Φ (∇.A) (b) Show that ∇.∇ Φ = ∇2Φ
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