CALC 3 EXAM 1 (10.1 - 10.4)

Find the equation of the sphere if one of its diameters has endpoints (−4,3,−4) and (−2,7,2).

(x+3)^2+(y−5)^2+(z+1)^2 − 14 = 0.

Find an equation of the largest sphere with center (4,10,6)(4,10,6) that is contained completely in the first octant.

(x−4)^2+(y−10)^2+(z−6)^2 − 16 = 0

Find an equation of the sphere that passes through the origin and whose center is (5,4,3)(5,4,3)

(x−5)^2+(y−4)^2+(z−3)^2 − 50 = 0

Determine if the pairs of vectors below are "parallel", "orthogonal", or "neither". 1. a=⟨5,5,−3⟩ and b=⟨−25,−25,−250/3⟩ are 2. a=⟨5,5,−3⟩ and b=⟨−25,−25,15⟩ are 3. a=⟨5,5,−3⟩ and b=⟨−20,−20,12⟩ are

1. a⋅b = 5(−25) +5(−25) −3(−250/3) = 0, so the vectors are orthogonal. 2. Because b=−5a, the vectors are parallel. 3. Because b=−4a, the vectors are parallel.

Find vectors that satisfy the given conditions: 1. The vector in the opposite direction to u = ⟨−2,1⟩ and of half its length is 2. The vector of length 9 and in the same direction as v =⟨5,0,2⟩

1. −1/2u = ⟨1,−0.5⟩ -> i - 0.5j 2. length of v = sqrt(29) 9⟨ 5/sqrt29 , 0/sqrt29 , 2/sqrt29 ⟩ = ⟨ 45/sqrt29 , 0/sqrt29 , 18/sqrt29 ⟩

Assume that u⋅v=6, ∥u∥=8, and ∥v∥=7. What is the value of 8u⋅(6u−3v)?

8u ⋅ (6u−3v) 48(u⋅u) − 24(u⋅v) 48∥u∥^2 − 24(u⋅v) Now substitute in our known values: 48(8^2)−24(6) = 2928

Find the unit vector in the direction opposite to v=⟨1,−4⟩

<-1/sqrt(17), 4/sqrt(17)>

Determine whether the vectors AB→ and PQ→ are equivalent. A=(−4,−4),B=(−8,−1),P=(−1,0),Q=(−5,3)

AB→=⟨−8+4,−1+4⟩=⟨−4,3⟩ PQ→=⟨−5+1,3−0⟩=⟨−4,3⟩ SO THEY ARE EQUIVALENT

You are looking down at a map. A vector u with |u| = 3 points north and a vector v with |v| = 5 points northeast. The cross product u×v points: A) south B) northwest C) up D) down Please enter the letter of the correct answer: The magnitude |u×v||u×v|:

By the right hand rule, the cross product points down. |u×v| = |u||v|sinθ = 3(5)sin(45degrees) = 7.5sqrt[2]

Find the area of the parallelogram with vertices: P(0,0,0), Q(−3,4,0), R(−3,2,−1), S(−6,6,−1).

Choose any three points from the given four. The area of the parallelogram is given by the magnitude of the cross product of two vectors built from those three points. For instance, choosing the points P,Q and R, we obtain the vectors PQ=⟨−3,4,0⟩ and PR=⟨−3,2,−1⟩ . The cross product of these two vectors is ⟨−4,−3,6⟩ with magnitude sqrt[(−4)^2+(−3)^2+(6)^2]= sqrt[61] Therefore the area of the parallelogram is sqrt[61]

Find the center and radius of the sphere x^2−14x+y^2−18y+z^2−8z=−82

Completing the squares in the equation gives (x^2−14x+49)+(y^2−18y+81)+(z^2−8z+16)= −82+49+81+16 ⇒(x−7)^2 + (y−9)^2 + (z−4)^2 = 64 which we recognize as an equation of a sphere with center (7,9,4)and radius 8.

Find two unit vectors orthogonal to a=⟨2,3,5⟩ and b=⟨−5,−1,4⟩ Enter your answer so that the first non-zero coordinate of the first vector is positive.

Go to 10.4 Problem 5 for explanations First Vector: ⟨ 17/sqrt[1547], -33/sqrt[1547], 13/sqrt[1547] ⟩ Second Vector: ⟨ -17/sqrt[1547], 33/sqrt[1547], -13/ sqrt[1547]⟩

Find the volume of the parallelepiped with adjacent edges PQ, PR, PS where P(2,5,−2),Q(4,8,1),R(1,4,−3),S(8,3,0).

Let a = PQ = ⟨2,3,3⟩, b = PR = ⟨−1,−1,−1⟩ and c = PS = ⟨6,−2,2⟩ cross product of these should equal 4 cubic units

Gandalf the Grey started in the Forest of Mirkwood at a point P with coordinates (1,−3) and arrived in the Iron Hills at the point Q with coordinates (3, 1). If he began walking in the direction of the vector v=3i+1j and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn?

Let w = PQ = ⟨2,4⟩ and let p = [v⋅w / |v|^2] ⋅ v = 10/10⟨3,1⟩ = ⟨3,1⟩ be the vector projection of w onto v. The point at which Gandalf the Grey changes direction is given by (1+3,−3+1) = (4,−2)

The nine Ring Wraiths want to fly from Barad-Dur to Rivendell. Rivendell is directly north of Barad-Dur. The Dark Tower reports that the wind is coming from the west at 62 miles per hour. In order to travel in a straight line, the Ring Wraiths decide to head northwest. At what speed should they fly (omit units)?

Since the Ring Wraights is flying northwest, we have v=⟨vcos⁡(135o),vsin⁡(135o) = ⟨−vsqrt2/2,vsqrt2/2⟩. Let w=⟨62,0⟩ be the wind's velocity vector. Since v+w must point north, it must be v(√2/2)=62. Thus v=62sqrt(2).

A child walks due east on the deck of a ship at 4 miles per hour.The ship is moving north at a speed of 5 miles per hour. Find the speed and direction of the child relative to the surface of the water.

Speed = abs(v) = sqrt(4^2 + 5^2) = √41 mph The angle of the direction from the north = arctan(4/5​) (radians)

A constant force F=−5i+0j+4k is applied to an object that is moving along a straight line from the point (3,−2,−1) to the point (−3,1,2). Find the work done if the distance is measured in meters and the force in newtons. Include units in your answer.

The displacement vector is D= (−3−3)i +(1+2)j +(2+1)k = −6i+3j+3k, so the work done is W = F ⋅ D = −5(−6)+0(3)+4(3)= 42 J

A horizontal clothesline is tied between 2 poles, 20 meters apart. When a mass of 2 kilograms is tied to the middle of the clothesline, it sags a distance of 2 meters. What is the magnitude of the tension on the ends of the clothesline? NOTE: Use g=9.8m/s2 for the gravitational acceleration.

The force due to gravity acting on the mass has magnitude 2Kg≈(2)(9.8)=19.6N, hence we have w=⟨0,−19.6⟩. The resultant T1+T2 of the tensile forces counterbalances w , so T1+T2=−w. Thus 2Tsinθ=19.6 and T=19.6/2sinθ =19.6/2sin(arctan(2/10)) =9.8/2 (sqrt(104))

Find the distance from (−2,7,−14) to each of the following: x-axis, y-axis, z-axis

The x-axis.Answer: √245 The y-axis.Answer: √200 The z-axis.Answer: √53

Find the cross product a×b where a=⟨−3,−3,1⟩ and b=⟨5,2,2⟩. a×b= Find the cross product c×d where c=2i+2j−5k and d=5i−3j+1k. c×d=

a x b = < -8, 11, 9 > c x d = < -13, -27, -16 >

Let a=⟨−2,0,−3⟩ and b=⟨1,−1,4⟩. Compute: a+b, a−b, 2a, 3a + 4b, abs(a)

a+b=⟨−2+1,0−1,−3+4⟩=⟨−1,−1,1⟩ a−b=⟨−2−1,0+1,−3−4⟩=⟨−3,1,−7⟩ 2a=⟨2(−2),2(0),2(−3)⟩=⟨−4,0,−6⟩ 3a+4b=⟨3(−2)+4(1),3(0)+4(−1),3(−3)+4(4)⟩=⟨−2,−4,7⟩ |a|= sqrt[(−2)^2+(0)^2+(−3)^2] = sqrt[13]

Find a⋅ b if |a| = 8 , |b| = 3, and the angle between a and b is π/7 radians.

a⋅b = |a| |b| cos(π/7) = 8(3)cos(π/7)

If a =⟨−5,−3,−3⟩ and b =⟨−2,−2,−3⟩, then a ⋅ b =? Is the angle between the vectors "acute", "obtuse" or "right"?

a⋅b = −5(−2) −3(−2) −3(−3) = 25 Since the dot product is positive, the angle between the vectors is acute.

I skipped 10.3 problem 17

do it

I skipped 10.3 problem 14 & 15

do them

Find the scalar and vector projection of the vector b=⟨0,−1,−1⟩ onto the vector a=⟨1,−5,5⟩. Scalar projection (i.e., component): Vector projection:

scalar projection: a⋅b / ∥a∥ = 0 / sqrt[51] vector projection: [a⋅b / ∥a∥^2] ⋅ a = (0/51)⟨1,−5,5⟩ = ⟨0,0,0⟩

Suppose u¯=⟨−3,−1⟩, and v¯=⟨8,6⟩ are two vectors that form the sides of a parallelogram. Then the lengths of the two diagonals of the parallelogram are

u + v = <−3,−1>+<8,6>=<5,5> AND u - v = <−3,−1>−<8,6>=<−11,−7> Their lengths are sqrt[(5)^2+(5)^2] = sqrt[50] AND sqrt[(−11)^2+(−7)^2]= sqrt[170]

Perform the following operations on the vectors u = <1,4,−2>, v =<2,−5,−5>, and w =⟨0,5,−4⟩: u ⋅ w (u ⋅ v) u ((w ⋅ w) u)u u ⋅ v + v ⋅ w

u ⋅ w =<1,4,−2>⋅<0,5,−4>=28 (u ⋅ v) u =−8<1,4,−2>=<−8,−32,16> ((w ⋅ w) u)u = (41<1,4,−2>) ⋅ <1,4,−2> = <41,164,−82> ⋅<1,4,−2>=861 u ⋅ v + v ⋅ w = −8−5 =−13

Let u¯=⟨1,0⟩u¯=⟨1,0⟩, v¯=⟨−3,2⟩, and w¯=⟨3,5⟩. Find the vector x¯ that satisfies: 7u¯− v¯+ x¯= 2x¯+w¯.

x¯= 7u¯−v¯− w¯= 7⟨1,0⟩ −⟨−3,2⟩ − ⟨3,5⟩ = ⟨7,−7⟩

What is the angle in radians between the vectors a=⟨7,−7,1⟩ and b=⟨4,1,−3⟩?

|a|= sqrt[(7)^2+(−7)^2+(1)^2] = sqrt[99], |b|= sqrt[(4)^2+(1)^2+(−3)^2] = sqrt[26], and a⋅b= 7(4)−7(1)+1(−3)=18. Then θ=arccos[(a⋅b)/(|a|⋅|b|)] = arccos[18/(sqrt[99]⋅sqrt[26])] radians

A rectangular box has length 17 inches, width 18 inches, and a height of 19 inches. Find the angle between the diagonal of the box and the diagonal of its base. The angle should be measured in radians.

θ=arccos[ ⟨17,18,19⟩⋅⟨17,18,0⟩ / |⟨17,18,19⟩|⋅|⟨17,18,0⟩| ] = arccos(sqrt[613] / sqrt[974])

Distance and Dot Products: Consider the vectors u=⟨2,6,5⟩ and v=⟨−3,4,9⟩. Compute ∥u∥ Compute ∥v∥ Compute u⋅v

‖u‖= sqrt[(2)^2+(6)^2+(5)^2]= sqrt[65] ‖v‖= sqrt[(−3)2+(4)2+(9)2] = sqrt[106] u⋅v = sqrt[ 2(−3)+6(4)+5(9) ] = 63

Let a=⟨−4,−2,−2⟩a=⟨−4,−2,−2⟩.Find a unit vector in the same direction as aa.

⟨ −4/√24​ ,−2/√24​ ,−2/√24​ ⟩

What is the terminal point of the vector a=⟨5,5⟩ based at P=(2,5)?

(7,10)

Find the equation of the sphere centered at (−3,5,−5) with radius 3.

(x+3)^2 + (y−5)^2 + (z+5)^2 − 9 = 0

Write down an (in)equality which describes the solid ball of radius 8 centered at (4,6,6). It should have a form like x^2+y^2+(z−2)^2−4 >= 0 where you use one of the following symbols ≤,<,=,≥,>.

(x−4)^2+(y−6)^2+(z−6)^2−64 ≤ 0

Find the cross product a×b where a=⟨1,4,1⟩ and b=⟨0,4,0⟩. a×b=

-4i + 0j + 4k OR < -4, 0, 4 >

What is the distance from the point (3,10,−5) to the xz-plane?

10

Section 10.2 Problem 15, Problem 16,

DO THEM

Determine whether the three points P=(5,8,−10), Q=(4,6,−13), R=(3,5,−16) are colinear by computing the distances between pairs of points.

Distance from P to Q: √14 Distance from Q to R: √11 Distance from P to R: 7 Are the three points colinear (yes/no)? no

A woman exerts a horizontal force of 7 pounds on a box as she pushes it up a ramp that is 7 feet long and inclined at an angle of 30 degrees above the horizontal. Find the work done on the box.

Here |D| = 7ft, |F|= 7 lb, and θ = 30 degrees. Thus W = F ⋅ D = |F|⋅|D| cosθ = 7(7)cos(30) = (49/2) ⋅ sqrt[3] ft-lb

Find the area of the triangle with vertices:Q(0,−2,4),R(−3,−3,3),S(−5,−5,4).

Let a=QR=⟨−3,−1,−1⟩ and b=QS=⟨−5,−3,0⟩. The area of the triangle is equal to half the length of the cross product of these two vectors. The cross product is given by a×b=⟨−3,5,4⟩, and it has magnitude |a×b| = sqrt[(−3)^2+(5)^2+(4)^2] = sqrt[50] Therefore the area of the triangle is (1/2)sqrt[50]

What are the projections of the point (−4,9,−4)(−4,9,−4) on the coordinate planes?

On the xy-plane: (−4 ,9 ,0 ) On the yz-plane: (0 ,9 ,−4 ) On the xz-plane: (−4 ,0 ,−4 )

Let R=(−5,2). Find the point P such that PR has components ⟨3,−3⟩

P=(-8,5)

If P=(3,−2) and Q=(4,−4), find the components of PQ

PQ=⟨4−3,−4+2⟩=⟨1,−2⟩

For what values of bb are the vectors ⟨−31,b,6⟩⟨−31,b,6⟩ and ⟨b,b2,b⟩⟨b,b2,b⟩ orthogonal?

The vectors are orthogonal if their dot product is zero. ⟨−31,b,6⟩ ⋅ ⟨b,b^2,b⟩ = −31b+b^3+6b = −25b+b^3 = b(b^2−25) Thus the vectors are orthogonal when b=0,5,−5.

Find a vector a that has the same direction as ⟨−8,7,8⟩ but has length 3

a = <-24/sqrt{177} , 21/sqrt{177}, 24/sqrt{177}>

10.4 problem 8 skipped

https://webwork.asu.edu/webwork2/Shaffer_MAT_267_Fall_2020/Section_10.4/8/?user=orios1&effectiveUser=orios1&key=kfzGNDVM8JEpiZ0Jz4wjiHQuKvmGdYGj&displayMode=MathJax&showOldAnswers=1

v=−2i+9j+10k w=−10j+4k Compute the dot product v⋅w

v⋅w = (−2i+9j+10k)⋅(−10j+4k) = (−2⋅0)+(9⋅−10)+(10⋅4) =−50

Let a=⟨4,4⟩ and b=⟨2,−1⟩.Show that there are scalars s and t so that sa+tb=⟨0,−12⟩

⇔s⟨4,4⟩+t⟨2,−1⟩=⟨0,−12⟩ ⇔⟨4s+2t,4s−1t⟩=⟨0,−12⟩ ⇔{4s+2t = 0 {4s−1t = −12 s = -2 t = 4