NAME CLASS PET519E

SURNAME HW NO 1 (Due

22/09/2014)

Signature

Student NO

CRN 14228

Instructor Dr. Mustafa Onur Attention! Make this page a cover page to your solutions. Show all your work for clearly and pricesely for full credit.

Given Date: 13/09/2014 (You should show all your work for clearly and pricesely for full credit.) Note: Throughout this Homework, a boldface letter will represent a vector.

Problem 1 (15 pts):

(a) (5 pts) By drawing a diagram show that if A + B = C, then B = C A. (b) (5 pts) Express C in terms of E, D, and F by considering Fig. 1.1 given below (you should give

me an equation by using addition and subtraction concepts for vectors. (c) (5 pts) Express G in terms of C, D, E, and B based on Fig. 1.1.

Fig. 1.1

Hint: In parts b and c, think of the vectors as displacements.

Problem 2 (15 points): Consider the two factors kjibkjia +=++= 322 and . Find compb a and projb a.

Problem 3 (20 points): Two vectors are arranged head to tail with an angle between their directions. Let BAC += , form CC and prove one form of the law of cosines:

2 2C= A +B +2ABcos() where A, B, and C represents the lengths of vectors CBA , and, , respectively. Actually, these lengths are scalars and equal to the l2 norm of the vectors, for example the length or l2 norm of the vector C is defined by

CCCC === 2C

Problem 4 (10 points): Find an equation of line passing through point (x0 = 2, y0 = 0, z0 = 4) and parallel to vector v = 2i + j + 3k, both in parametric and nonparametric form.

A

BC

C

D

E

F

G B

NAME CLASS PET519E

SURNAME HW NO 1 (Due

22/09/2014)

Signature

Student NO

CRN 14228

Instructor Dr. Mustafa Onur Attention! Make this page a cover page to your solutions. Show all your work for clearly and pricesely for full credit.

Problem 5 (5 pts): Find an equation of a plane passing through the point (x0 = 1, y0 = 3, and z0 = -6) and perpendicular to the vector n = 3i - 2j + 7k. Problem 6 (10 points): Find the gradients of the following functions f, i.e., find grad f or ),( yxf (5 pts) (a) zexzyxf xy lnsin),,( ++= (5 pts) (b) r/1),,( =zyxf , where r is the position vector. Problem 7 (15 points):

(5 pts) (a) Find a unit vector normal to the isotimic surface x2 + yz = 5 at the point (2,1,1). (10 pts) (b) Find the directional derivative df/ds at (1,3,-2) in the direction of a = i + 2j + 2k if

r

1),,( =zyxf , where r is the position vector and also interpret if f is increasing or

decreasing in that direction.

Problem 8 (10 points): (a) (5 pts) Find the divergence of the vector field F = x2 i + y2 j + z2 k and then (b) (5 pts) the find Laplacian of the resulting function from the divergence of the given field.