Engineering Mathematics I Tutorial Sheet 5 PDF

Jaypee University of Information Technology Department of Mathematics B. Tech. Ist Semester Engineering Mathematics I (24B11MA111) Tutorial Sheet 5 1. Find the vector and Cartesian forms of the equation of line passing...

Jaypee University of Information Technology Department of Mathematics B. Tech. Ist Semester Engineering Mathematics I (24B11MA111) Tutorial Sheet 5 1. Find the vector and Cartesian forms of the equation of line passing through (i) the origin and the point (5,-2, 3). (ii) the point P(3,-4,-1) and parallel to the vector 𝚤̂ + 𝚥̂ + 𝑘. (iii) the point (1, 1, 1) and parallel to the z-axis. (iv) the point (2, 4, 5) and perpendicular to the plane 3𝑥 + 7𝑦 − 5𝑧 = 21. (v) the point (2,3,0) and perpendicular to 𝐴⃗ = 𝚤̂ + 2𝚥̂ + 3𝑘 and 𝐵⃗ = 3𝚤̂ + 4𝚥̂ + 5𝑘 2. Find the equation for the plane passing through: (i) the points (1,3,2), (3, −1, 6), (5,2,0). (ii) the point 𝑃 (0,2, −1) & normal to 𝑛⃗ = 3𝚤̂ − 2𝚥̂ − 𝑘. 3. Find the point of intersection of the lines 𝑥 = 2𝑡 + 1, 𝑦 = 3𝑡 + 2, 𝑧 = 4𝑡 + 3 and 𝑥 = 𝜆 + 2, 𝑦 = 2𝜆 + 4, 𝑧 = −4𝜆 − 1. Also find the plane determined by these lines. 4. Find the length of the curve defined by 𝑟⃗(𝑡) = 6𝑡, 3√2𝑡 , 2𝑡 , 0 ≤ 𝑡 ≤ 1. 5. Find the directional derivative of the functions: (i) 𝑓(𝑥, 𝑦, 𝑧) = 2𝑦𝑧 + 𝑧 in the direction of the vector 𝚤̂ + 2𝚥̂ + 2𝑘 at the point(1, −1, 3). (ii) 𝑓(𝑥, 𝑦, 𝑧) = 𝑥 + 𝑦 + 𝑧 at the point (1, −1, 2) in the direction of 𝚤̂ + 2𝚥̂ + 𝑘. (iii) 𝑓(𝑥, 𝑦, 𝑧) = 𝑥𝑧 − 3𝑥𝑦 + 2𝑥𝑦𝑧 − 3𝑥 + 5𝑦 − 17 from the point 𝑃(2, −6, 3) in the direction of origin O. 6. Find the equation of the tangent plane and the normal line at the point P of the given surfaces: (i) 𝑥 𝑦 + 𝑥𝑧 = 2𝑦 𝑧, 𝑃 = (1,1,1). (ii) 2𝑧 − 𝑥 , 𝑃 = (1,1,1). (iii) 4 − 𝑥 − 2𝑦 , 𝑃 = (1, −1,1). 7. Sketch the curve 𝑓(𝑥, 𝑦) = 𝑐 together with ∇𝑓 and the tangent line at the given point. Also determine the equation of the tangent line. (i) 𝑥 + 𝑦 = 4, (√2, √2 ). (ii) 𝑥𝑦 = −4, (2, −2). 8. Answer the following: (i) Find the gradient of 𝑓(𝑥, 𝑦) = 𝑦 − 4𝑥𝑦, at (1,2). (ii) Find the gradient of 𝑓(𝑥, 𝑦, 𝑧) = 𝑥 𝑦 + 𝑥𝑦 − 𝑧 , at (3, 1, 1). (iii) For a force field 𝐹⃗(𝑥, 𝑦) = 𝚤̂ − 𝚥̂ applied on an object, find 𝑓(𝑥, 𝑦) such that 𝑔𝑟𝑎𝑑 𝑓 = 𝐹⃗(𝑥, 𝑦). ⃗ 9. If 𝑟⃗ = 𝑥𝚤̂ + 𝑦𝚥̂ + 𝑧𝑘, |𝑟⃗| = 𝑟 and 𝑟 = , show that 𝑔𝑟𝑎𝑑 = −. 10. Evaluate 𝑑𝑖𝑣 𝑐𝑢𝑟𝑙 𝐹⃗ where𝐹⃗ = 𝑦𝑧 𝚤̂ + 𝑥𝑦𝚥̂ + 𝑦𝑧𝑘. *** Answers 1. (i) 𝑟⃗ = 5𝑡 𝚤̂ − 2𝑡𝚥̂ + 3𝑡𝑘 ; x=2t, y= -2t, z=3t (ii) 𝑟⃗ = (3 + 𝑡) 𝚤̂ − (𝑡 − 4)𝚥̂ + (𝑡 − 1)𝑘; x=3+t, y= t-4, z= t-1 (iii) 𝑟⃗ = 𝚤̂ + 𝚥̂ + (𝑡 + 1)𝑘; x = 1, y = 1, z = 1+t (iv) 𝑟⃗ = (2 + 3𝑡) 𝚤̂ − (4 + 7𝑡)𝚥̂ + (5 − 5𝑡)𝑘; x = 2+3t, y = 4+7t, z = 5-5t (v) 𝑟⃗ = (2 − 2𝑡) 𝚤̂ − (3 + 4𝑡)𝚥̂ + (−2𝑡)𝑘; x = 2 – 2t, y = 3 + 4t, z = -2t 2. (i) 6𝑥 + 10𝑦 + 7𝑧 = 50 (ii) 3𝑥 − 2𝑦 − 𝑧 = −3 3. −20𝑥 + 12𝑦 + 𝑧 = 7 4. 8 √ 5. (i) (ii) (iii) 18 6. (i) 𝑥 − 𝑦 = 0, (1 + 3𝑡, 1 − 3𝑡, 1) (ii) 2𝑥 − 𝑧 − 2 = 0, (2 − 4𝑡, 0, 2 + 2𝑡) (iii) 𝑥 − 2𝑦 − 3 = 0, (1 − 𝑡, −1 + 2𝑡, 1) 7. (i) 2√2, 2√2 , 𝑦 = −𝑥 + 2√2 (ii) (−2,2), 𝑥 − 𝑦 = 4 8. (i) −8 𝚤̂ (ii) 7 𝚤̂ + 24𝚥̂ − 2𝑘 (iii) 𝑓(𝑥, 𝑦) = − + 𝑘; 𝑘 𝑖𝑠 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡. 9. N.A. 10. 0