Question

Let u = <-4, 1>, v = <-1, 6>. Find -2u + 4v.

Answer 1

@agent0smith

Answer 2

u = <-4, 1> is given

to find -2u, you multiply everything in vector u by -2

so,

u = <-4, 1>

-2u = -2<-4, 1>

-2u = <-2(-4), -2(1)>

-2u = <8, -2>

Answer 3

If v = <-1, 6>, then 4v = ???

Answer 4

<4, 22>

<4, 7>

<12, -26>

<10, -14>

Answer 5

Do you see how I got -2u?

Answer 6

so it would be <4, 22> ? @jim_thompson5910

Answer 7

wait no

Answer 8

If v = <-1, 6>, then 4v = ???

Answer 9

<12, -26>

Answer 10

to calculate 4v, multiply everything in v by 4

Answer 11

yes i did

Answer 12

@jim_thompson5910 then i added them and got
<12, -26>

Answer 13

tell me what you got for 4v

Answer 14

-4,24

Answer 15

4,22

Answer 16

so

-2u + 4v

turns into

<8, -2> + <-4,24>

Answer 17

Then you add up the corresponding components

8 + (-4) = 4
-2 + 24 = 22

So yes, it's <4, 22>

Answer 18

thank you for explaining c:

Answer 19

you're welcome

Answer 20

@jim_thompson5910 do you know how to explain this by any chance?
Find the angle between the given vectors to the nearest tenth of a degree.

u = <8, 4>, v = <9, -9>

Answer 21

are you familiar with dot products?

Answer 22

no :(

Answer 23

it turns out that if

u = <a,b>
v = <c,d>

then

u dot v = a*c + b*d

Answer 24

so you multiply the corresponding entries, then add up those products

Answer 25

if u = <8, 4>, v = <9, -9>, then u dot v = ??

Answer 26

8*9 + 4*-9
72+ -36

Answer 27

keep going

Answer 28

36

Answer 29

but my answer choices are :
81.6°

25.8°

35.8°

71.6°


so would it be 35.8?
@jim_thompson5910

Answer 30

ok so that's part of finding the angle

Answer 31

idk, we haven't finished yet

Answer 32

now we must find the lengths of vector u and vector v

Answer 33

oh okay sorry c:

Answer 34

if u = <8, 4>, then |u| = ??

Answer 35

|u| is the length of vector u

Answer 36

-8 -4?

Answer 37

|u| = sqrt( u dot u )

Answer 38

so you first compute the dot product u dot u
then take the square root

Answer 39

wow you lost me.. im so sorry i really suck at this

Answer 40

@jim_thompson5910

Answer 41

u is a vector, so it has a length and direction

Answer 42

the length of vector u is found by first computing the dot product of u dot u

what is that dot product?

Answer 43

I'm guessing you're stuck on how to find u dot u?

Answer 44

yes,,

Answer 45

did you see how to find u dot v?

Answer 46

thats the one that i got 36 for right?

Answer 47

yes

Answer 48

you would do the same, but instead of use v, you use vector u

Answer 49

so you just use the same vector twice

Answer 50

162?

Answer 51

I'll show you what I mean

u = <8, 4>

u dot u = 8*8 + 4*4

u dot u = 64 + 16

u dot u = 80

Answer 52

or another way to put it

<8, 4> dot <8,4> = 8*8 + 4*4 = 80

Answer 53

So the length of vector u is sqrt(80) units long

Answer 54

Using this idea, how long is vector v?

Answer 55

81.6°

Answer 56

how long is vector v?

Answer 57

or, tell me what <9, -9> dot <9, -9> is equal to

Answer 58

162

Answer 59

good, so the length of vector v is sqrt(162)

Answer 60

we now multiply the two lengths to get


sqrt(80)*sqrt(162)

Answer 61

We'll then divide the dot product u dot v by sqrt(80)*sqrt(162)

and then finally, take the arccosine of that result

Answer 62

so

36/( sqrt(80)*sqrt(162) )

then the arccosine of that

Answer 63

0.31622776601

Answer 64

now take the arccosine of that

Answer 65

make sure you are in degree mode

Answer 66

1.24904577

Answer 67

seems like you're not in degree mode

Answer 68

i dont really have a calculator so im using googles

Answer 69

ok, then you would type in

"arccos(0.31622776601) in degrees"

type that without quotes

Answer 70

you have to stick on "in degrees" because google uses radians by default

Answer 71

71.6

Answer 72

what decimal number did you get (before you rounded)

Answer 73

71.5650512

Answer 74

very good, so 71.6 is correct

Answer 75

thank you!

Answer 76

to sum things up, the angle theta is found by using the formula

\[\Large \theta = \arccos\left(\frac{u \cdot v}{|u||v|}\right)\]