Pre Cal Unit 9

The direction (slope) of the vector is _____. v=initial point (−4,1), terminal point (−2,−1)

-1 1-(-1) 2 ----- = ---- = 1 -4-(-2) -2

The direction (slope) of the vector is _____. u=initial point (4,−1),(4,-1), terminal point (−1,4)

-1 4-(-1) 5 ------- = ----- = -1 -1-4 -5

Let u = -5i - 7j and v = -3i + 5j Find u + 2v

-11i + 3j

Given: u=-i+3j, v=8j, w=5i-4j Find the answer: u·w

-17

Let u = -5i - 7j and v = -3i + 5j Find u - v

-2i - 12j

Let u = -i - 5j and v = -2i - 6j Find u+v2

-2i - 8j

Given: u=-i+3j, v=8j, w=5i-4j Find the answer: v·w

-32

Let u = -i - 5j and v = -2i - 6j Find 2v - u

-3i - 7j

Let u= ⟨4,−7⟩, v= ⟨1,2⟩, w= ⟨8,3⟩, c= 55 Find c·(u·v)

-50

Let u = i + 3j and v = -4i - 2j Find u + 2v.

-7i - j

A unit vector has a magnitude of __________.

1

The direction (slope) of the vector is _____. u=initial point (5,−2), terminal point (0,3)

1

Find the magnitude of the vector. u= ⟨6,−8⟩

10

Given u=⟨4,−7⟩, v=⟨1,2⟩, w=⟨8,3⟩, c=5 Find v⋅w

14

Let u= ⟨4,−7⟩, v= ⟨1,2⟩, w= ⟨8,3⟩, c= 55 Find v·w

14

Let u= ⟨4,−7⟩, v= ⟨1,2⟩, w= ⟨8,3⟩, c= 5 Find c·(v·v)

25

Given: u=⟨3,−3⟩, v=⟨5,2⟩, w=⟨6,4⟩, c=5 Find v⋅w

38

Let u = i + 3j and v = -4i - 2j Find -u - v.

3i - j

Find the magnitude of the vector. r=⟨2,√12⟩

4

Solve: Find the amount of work done. A force of 52 pounds is applied at an angle of 60° above a the horizontal to slide a table 25 feet across a floor.

W=650 foot-pounds

The symbol for the standard vertical unit vector is _______.

j

resolve

separate into basic parts

Given: u= ⟨0,−2⟩ , v= ⟨−7,4⟩ , w= ⟨5,8⟩ Find the answer: u·v

-8

Find the magnitude of the vector. u= ⟨2, √12⟩

4

The direction (slope) of the vector is _____. u=initial point (3, 1), terminal point (−4, −3)

4/7

Given: u=⟨3,−3⟩, v=⟨5,2⟩, w=⟨6,4⟩, c=5 Find c(u⋅v)

45

Find the magnitude of the vector. u= ⟨−4,3⟩

5

Find the magnitude of the vector. w=⟨3,4⟩

5

The magnitude of the vector is _____. u=initial point (4,−1), terminal point (−1,4)

5 ⋅ √2

Let u = i + 3j and v = -4i - 2j Find 2u + v/2

5j

Find the magnitude of the vector. v=⟨5, √24⟩

7

Find the magnitude of the vector. u= ⟨5,√39⟩

8

Find the magnitude of the vector. v=⟨4, √48⟩

8

Find the angle between the vectors to two decimal places. Use a calculator. u=⟨6,3⟩, v=⟨2,−4⟩

90°

Find the angle between the vectors to the nearest whole number. Use a calculator. v=⟨−2,3⟩, w=⟨3,2⟩

=1.5°

Find the angle between the vectors to one decimal place. Use a calculator. u=−5j, v=3i

=90.0°

Solve. Tyrone is pulling his sister in a wagon with a force of 30 pounds on a handle that makes an angle of 25° with the horizontal. How much work is done if he pulls the wagon 55 feet? (to the nearest whole number)

W≈1,495 foot-pounds

The sixth step to finding the angle between two vectors is: Find the __________________ of the quotient to obtain the angle.

arcosine

magnitude

quantity, size, amount

Solve: A truck weighing 17,000 pounds is parked on a hill with a 12° slope. Find the force needed to prevent the truck from rolling down the hill. (to the nearest whole number)

|F1|=3,534

Solve:Earl has to hold a 1,240 pound piano on a ramp that is slanted 20° from the horizon. How much force must he exert to keep the piano from moving? (to the nearest whole number)

|F1|=424

Solve: How much force would be needed to prevent a 30,000 pound truck from rolling down a hill with a slope of 10°?

|F1|=5,208| pounds

The magnitude of the vector is _____. u=initial point (1,1), terminal point (4,3)

√13

Given: u=⟨3,−3⟩, v=⟨5,2⟩, w=⟨6,4⟩, c=5 Find u⋅(v+w)

15

Find the components of the resultant vector u+v. _________________. Initial Point (x1,y1) Terminal Point (x2, y2) u (6, −2) (0,1) v (2, 2) (−2, −2)

u + v = ⟨−10, −1⟩

Find the unit vector u in the direction of v. v = ⟨10,√21⟩

u = ⟨10/11,√21/11⟩

Find the unit vector u in the direction of v. Vector v has an initial point (0,3) and a terminal point (1,4)

u = ⟨−1/√2,−1/√2⟩

Find the unit vector u in the direction of v. Vector v has an initial point (0,−1) and a terminal point (−3,4)

u = ⟨−3√34 ,5/√34⟩

Find the component vectors. v=⟨3i−5j⟩ onto w=⟨6i+2j⟩ v1= v2=

v1=⟨(6/5)i+(2/5)j⟩ v2=⟨(9/5)i−(27/5)j⟩

A ___________________ represents an object with both magnitude and direction.

vector

Find the vector 2v+w. v = ⟨−4, 5⟩ , w = ⟨1, 6⟩

⟨−7,16⟩

Let u = -i - 5j and v = -2i - 6j Find -u + v

-i - j

Given: u=cos30°i+sin30°j, v=cos60°i-sin60°j Find the answer: u·v

0

Are the following vectors equivalent? u=initial point (0,0), terminal point (−3,−4) v=initial point (4,3), terminal point (0,0)

No

orthogonal

descriptive of meeting at right angles

Find the components of the vector v. Use a calculator. Answer to 2 decimal places. magnitude=12; direction angle =300°

≈≈⟨6.00,−10.44⟩〈6.00,-10.44〉

Find the unit vector u in the direction of v. v = ⟨10 , √21⟩

⟨10/11,√21/11⟩

Let u= ⟨4,− 7⟩, v= ⟨1, 2⟩, w= ⟨8, 3⟩, c = 22 Find c · (v+w)

⟨18,10⟩

Find the vector 2(v-w). v = ⟨−4, 5⟩ , w = ⟨1, 6⟩

⟨−10, −2⟩

Find the vector 4w+2v. v = ⟨−4, 5⟩ , w = ⟨1, 6⟩

⟨−4, 34⟩

Find the component vectors. r=⟨−2,3⟩r=〈-2,3〉 onto s=⟨6,4⟩s=〈6,4〉 r1= r2=

. r1=⟨0,0⟩ r2=⟨−2,3⟩

Find the angle between the vectors to the nearest whole number. Use a calculator. u= cos30°i + sin30°j, v = cos40°i + sin40°j

180.0°

The direction (slope) of the vector is _____. v=initial point (0,−2),(0,-2), terminal point (3,0)

2 ---- 3 0-(-2) 2 ------- = ---- 3-0 3

The magnitude of the vector is _____. v=initial point (−4,1), terminal point (−2,−1)

2 ⋅ √2

The direction (slope) of the vector is _____. u=initial point (1, 1), terminal point (4,3)

2/3 3-1 2 ----- = ---- 4-1 3

Two vector operations are vector addition and scalar _____________________.

Multiplication

Solve: The horse is pulling the plow with a harness attached to the plow at an angle 10° from horizontal. How much work would be done if a force of 90 pounds was exerted over 1,248 feet. ( Round to the nearest whole number)

W≈110,613 foot-pounds

Find the unit vector u in the direction of v. v = ⟨−4,3⟩

u = ⟨−4/5, 3/5⟩

Find the unit vector u in the direction of v. v = ⟨−4,7⟩

u = ⟨−4/√65, 7/√65⟩

Find the unit vector u in the direction of v. Vector v has an initial point (3,-2) and a terminal point (−2,4)

u = ⟨−5 / √61, 6 / √61⟩

The magnitude of the vector is _____. v=initial point (0,−2),(0,-2), terminal point (3,0)

√13

Find the magnitude of the vector v. initial point = (1,3) terminal point = (−1,−2)

√29

Find the angle between the vectors to two decimal places. Use a calculator. u=2i−3j, v=i−j

≈≈ 11.30°

Find the angle between the vectors to one decimal place. Use a calculator. u=⟨4,4⟩, v=⟨2,3⟩

≈≈ 11.3°

Vectors that are equivalent have the same magnitude and _____.

direction

The fifth step to finding the angle between two vectors is: _______________ the dot product by the square root of the answer.

divide

Select the correct word for the blank: The ____________________ of two vectors is a scalar.

dot product

The standard unit vector i is a __________ vector.

horizontal

The symbol for the standard horizontal unit vector is _______.

i

The ___________________ of vector v is represented by |v|.

magnitude

The third step to finding the angle between two vectors is:_________________ the sums of the magnitudes times each other

multiply

Are the following vectors equivalent? u=initial point (3, −1), terminal point (−1,2) v=initial point (−1, 3), terminal point (2,−1)

no

component

one of two or more scalar quantities used to indicate the size or magnitude of a vector

Select the correct word for the blank: If the dot product of two vectors is 0, they are ________________.

orthogonal

Write the vector as a linear combination of the unit vectors i and j. r = ⟨−2,7⟩

r = -2i + 7j

The fourth step to finding the angle between two vectors is: Take the square ___________ of the answer.

root

Applications that have magnitude but not direction are _____.

scalars

The second step to finding the angle between two vectors is: Find the ________ of the ____________ of the x- and y-components of each magnitude.

squares sum

resultant

sum of the vectors

Find the components of the resultant vector u+v _________________. Initial Point (x1,y1) Terminal Point (x2,y2) u (−5,−3) (−3, 1) v (5,2) (4, −5)

u + v = ⟨1, −3⟩

Find the magnitude of the vector. u=⟨−3,3⟩

√18

Find the direction of the angle θθ to the nearest degree. Use a calculator. v= ⟨5,−2⟩

≈≈ 338°

Find the angle between the vectors to one decimal place. Use a calculator. u=cos45°i+sin45°j, v=cos90°i+sin90°j

≈≈ 45.0°

Solve: Lin Li's dad is pulling his son and dog on a toboggan. He is pulling on a rope 37° from horizontal with a force of 57 pounds. How much work would he do if he pulled the toboggan 102 feet? (to the nearest tenth)

W≈4643.1 foot-pounds

Are the following vectors equivalent? u=initial point (1,1), terminal point (4,7) v=initial point (−2,2), terminal point (1,8)

Yes

What two formulas would you use to determine if two vectors are equivalent?

Point-Slope Distance

Find the unit vector u in the direction of v. v = ⟨−6 ,√13⟩

u = ⟨−6/7, √13, 7⟩

Find the component vectors. u=⟨−3,−3⟩ onto v=⟨2,6⟩ u1= u2=

u1 = ⟨−6/5,−18/5⟩ u2 = ⟨−9/5,3/5⟩

Find the component vectors. u=⟨4,−2⟩ onto v=⟨5,3⟩ u1= u2=

u1= ⟨35/17 ,21/17⟩ u2= ⟨33/4 ,−55/4⟩

Find the component vectors. u=⟨1,2⟩ onto v=⟨3,1⟩ u1= u2=

u1=⟨3/2,1/2⟩ u2=⟨−1/2,3/2⟩

A __________ vector has a magnitude of one.

unit

The standard unit vector j is a __________ vector.

vertical

Given; u=−5i−7j, v=−3i+5j Find the answer: u+2v

−11i+3j

Given: u=−5i−7j, v=−3i+5j Find the answer: u−v

−2i−12j

Find the magnitude of the resultant vector u+v _________________. Initial Point (x1,y1) Terminal Point (x2,y2) u (−5, −3) (−3, 1) v (5, 2) (4, −5)

√10

Find the magnitude of the resultant vector u+v _________________. Initial Point (x1, y1) Terminal Point (x2, y2) u (6,−2) (0,1) v (2,2) (−2,−2)

√101

Find the direction of the angle θθ to the nearest degree. Use a calculator. v= ⟨6,7⟩

≈≈ 49°

Find the angle between the vectors to two decimal places. Use a calculator. u=⟨2,5⟩, v=⟨−2,1⟩

≈≈ 85.24°

Solve: A force of 1,000 pounds is applied along WZ−→ from W to Z at an angle of 27° from the horizon. WZ−→−= 7i+−−√15j . How much work was done?

W≈7,128 foot-pounds

Write the vector as a linear combination of the unit vectors i and j. v = ⟨4,−8⟩

v = 4i - 8j

Find the direction of the angle θθ to the nearest degree. Use a calculator. v= ⟨−3,−4⟩

≈≈ 233°

Find the components of the vector v. Use a calculator. Answer to 2 decimal places. magnitude=3; direction angle =283°

≈≈ ⟨0.68,−2.92⟩

Find the components of the vector v. Use a calculator. Answer to 2 decimal places. magnitude=5; direction angle =27°

≈≈ ⟨4.45,2.25⟩

Find the components of the vector v. Use a calculator. Answer to 2 decimal places. magnitude=7; direction angle =98°

≈≈ ⟨−0.97,6.93⟩

Write the vector as a linear combination of the unit vectors i and j. v = ⟨1,−5⟩

v = i - 5j

Find the components of the vector v. initial point = (1,3), terminal point = (−1,−2)

v = ⟨−2,−5v⟩ Find it's length in the xx and yy direction. −1 − 1 = −2 −2 − 3 = −5

Find the component vectors. v=i−j onto w=2i+5j v1= v2=

v1= −6/29i − 15/29j v2= 35/29i − 14/29j

Find the component vectors. v=⟨i−2j⟩v=〈i-2j〉 onto w=⟨2i+4j⟩ v1= v2=

v1= ⟨(−35)i−(65)j⟩ v2=⟨(85)i−(45)j⟩

Write the vector as a linear combination of the unit vectors i and j. Vector w has an initial point (1,6) and a terminal point (−2,1).

w= -3i - 5j

Write the vector as a linear combination of the unit vectors i and j. Vector z has an initial point (0,9) and a terminal point (2,−3).

z = 2i - 12j

Write the vector as a linear combination of the unit vectors i and j. Vector z has an initial point (-7,2) and a terminal point (3,-5).

z = 10i - 7j

Solve: Marvin think he is ready to ski down a hill with a 45° incline.With all his equipment, he weighs 145 pounds. How much force (F1) would it take to hold Marvin from going down the hill at the starting gate? (Round to the nearest whole number)

|F1|=103pounds

Solve: How much force is required to hold a 300 pound snowball from rolling down a hill that has a slope of 45°. (to the nearest whole number)

|F1|=212

Solve: How much force must be exerted to stop Manny's car from rolling down a hill with a slope of 30°? The weight of the car and Manny is 120 pounds.

|F1|=60 pounds

Given: u= ⟨3,3⟩, v= ⟨1,8⟩, w= ⟨0,2⟩; Prove: |v|^2=?v⋅v

|⟨1,8⟩|^2=?⟨1,8⟩⋅⟨1,8⟩| (√1^2 + 8^2)^2 =? (1×1)+(8×8) (√1+64)^2 =? 1+64 (√65)^2 =? 65 65 = 65

Find the direction of the angle θθ to the nearest degree. Use a calculator. v= ⟨−3,2⟩

≈≈ 146°

Given: u= ⟨0,−2⟩ ,v= ⟨−7,4⟩ , w= ⟨5,8⟩ Find the answer: (u·v)w

⟨−40, −64⟩

The first step to finding the angle between two vectors is: Add the product of the _____ of the two vectors to the product of the ______ of the two vectors.

x-values y-values

Are the following vectors equivalent? u=initial point (1,2), terminal point (3,4) v=initial point (4,4), terminal point (6,6)

yes

What two formulas would you use to determine if two vectors are equivalent?

Distance Point-Slope

scalar

a real number having magnitude but no direction

projection

an image or representation of a vector on another vector

Find the components of the vector v. Use a calculator. Answer to 2 decimal places. magnitude=8; direction angle =172°

≈≈ ⟨−7.92,1.11⟩