(MATH101)problem_answer.pdf

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MATH 101 Problem Sets

Set 1

1. Check if a . b

(a)

a = i +2j . k, b = .2i . 4j +2k

(b)

a = i + j, b = j + k

2. Find the dot product and the angle between the two vectors

a = i + j . k, b =2i . 3j +4k

3. Determine whether PQ and PR are perpendicular

P =(.1, 3, 0),Q =(2, 0, 1),R =(.1, 1, .6).

4.

Find a b and c (a b)

a = i + j + k, b = i . k, c = .i + j . k.

5.

Let u and v be adjacent sides of a paralelogram. Use vectors to show that the parallegram is a rectangle if the diagonals are equal in length.

6.

Consider the two-dimensional xy-plane. Let O be the origin and A be a point lying on the x-axis. Let P be an arbitrary point of a curve. If the angle .

OPA (P being the vertex) is always a right angle, what is the geometical object traced by P (the curve)?

1.

What is the area of the triangle which has vertices at (2, 1, 2), (3, 3, 3), (5, 1, 2)?

2.

Find a vector equation, and the parametric equations for the line that contains the point (.2, 1, 0), and is parallel to the vector 3i . 2j + k.

3.

Find an equation of the plane that contains the point (.1, 1, 3) and has normal

k.

vector . 2i +15j . 1

2

4.

Find an equation of the plane that contains the points (2, .1, 4), (5, 2, 5), and (2, 1, 3).

5.

Show that the vector ai + bj + ck is a normal to the plane ax + by + cz = d where a, b, c, d are constants.

6.

Use vectors to show that for any triangle the three lines drawn from each vertex to the midpoint of the opposite side all pass through the same point.

1. Determine the component functions and domain of the given function

(a) F= t +1i + 1 . tj + k

(b) F(t)=(ti + j) (ln(t)j +1k)

t

2. Sketch the curve traced out by the vector-valued function. Indicate the direction in which the curve is traced out.

(a) F(t)= ti + tj + tk

(b) F(t)= cos ti +sin tj + tk

3. Compute the limits or explain why it does not exist sin t (e.t . 1)

(a) lim ( i +(t + 2)j + k)

t0tt

(b) limF(t)where

t0

.

. .1/t22

i + ek =0

. tj + tfor t

F (t)= 1

.

. j for t =0

2

4. Find the derivatives of the function

(a)

F(t)= t2 cos ti + t3 sin tj + t4k

(b)

F Gwhere

F(t)= (1+ t2)i +(et + e.t)j, 11

G(t)= i + j

t + t 2+ sin t

(c) f(t)F(t)where f = t2 , F= i +1j +1 k

tt2

11

(d) F . f where f = t2 ,F = i + j + k

tt2

5. Find the velocity, speed, acceleration, and magnitude of acceleration of an object having the given position function

r(t)=cos t i +sin t j + tk

1. Give the domain of de.nition for which the function is de.ned and real, and indicate

this domain graphically

(a)

f(x, y)= 1 . x2 . y2

(b)

f(x, y)= ln(x + y)

2. Sketch the level curve f(x, y)= c for

(a)

f(x, y)= 1 . x2 . y2; c =0, 1/ 2

(b)

f(x, y)= x2 . y2; c = .1, 0, 1

3. Sketch the graph of f

(a)

f(x, y)= 1 . x2 . y2 (an ellipsoid)

(b)

f(x, y)= x2 + y2 (a cone)

(c)

f(x,