Question

PLEASE HELP: Find vectors v and w, such that v is parallel to (1,2,3), v + w = (7,3,5) and w is orthogonal (perpendicular) to (1,2,3).

Answer 1

@infinitylove, if two vectors are parallel then they are scalar multiples of each other. If two vectors are orthogonal then their dot product equals zero. Is that helpful? If yes then try it and I will check your answer to see if you've done it correctly.

Answer 2

@calculusfunctions, Thank you for your help. I understand both of those facts and I've been working with them both, but they haven't gotten me very far. From this information, I know:

(v, v, v) + (w,w,w) = (7,3,5)
(1,2,3) dot (w,w,w) = 0
And, like you said: if two vectors are parallel then they are scalar multiples of each other.

Can you help further?

Answer 3

@calculusfunctions Can you help me find v and w?

Answer 4

This one is somewhat tricky to explain in this format. I'm thinking.

Answer 5

Are you sure you've written the question exactly? Could you please double check?

Answer 6

COPIED AND PASTED FROM MY TEXTBOOK: Find vectors v and w such that v is parallel to (1; 2; 3), v + w = (7; 3; 5) and w is orthogonal to
(1; 2; 3).

Answer 7

@calculusfunctions

Answer 8

I understand if you can't help. These questions are challenging. Do yo think you could help me with another? @calculusfunctions

Answer 9

Sure but it's just that it's my nature that this will now bother me until I'm able to explain it to you. lol. Give me a few more minutes. Thanks.

Answer 10

Haha I totally understand. I'm a math major in college and hit some challenging questions in linear algebra. Take your time! Thanks a lot :) @calculusfunctions

Answer 11

Thanks. Don't worry I'm right here. If I'm silent is because I'm thinking and working out the problem. Thanks for being patient.

Answer 12

Eureka!! Are you there?

Answer 13

YES! haha @calculusfunctions

Answer 14

@infinitylove, OK, first let vector v = (k, 2k, 3k) and let vector w = (a , b, c). If

v + w = (7, 3, 5) then

k + a = 7, 2k + b = 3, and 3k + c = 5.

Now first solve each of these equations for a, b, and c. Hence

a = 7 − k, b = 3 − 2k, and c = 5 − 3k.

Please tell me you understand thus far before we continue.

Answer 15

Remember that k is a constant.

Answer 16

So now @infinitylove, you know that vector w = (7 − k, 3 − 2k, 5 − 3k). Right?

Answer 17

YUP! haha @calculusfunctions

Answer 18

So do you know how to get the answer now? Is this a Eureka! moment for you also?

Answer 19

@infinitylove, I'm anxiously waiting to know if you arrived at the correct answer.

Answer 20

Haha, okay, I understand everything we've done so far. So now we know: v = (k, 2k, 3k) and w =(7-k, 3-2k, 5-3k), correct?

Answer 21

Yes!! Now do you know what to do next?

Answer 22

So (k,2k,3k) + (7-k,3-2k,5-3k) = (7,3,5)

Answer 23

Do I make individual equations to find the value of k? Like, k + 7-k = 7, but the k's cancel out. And 2k + 3-2k = 3, but the 2k's cancel out. And 3k + 5-3k = 5, but the 3k's cancel out. I'm a little mixed up. How do we solve for k?

Answer 24

Well you want to make sure that w is orthogonal to (1, 2, 3) so I would do\[(1, 2, 3)∙(7-k, 3-2k, 5-3k)=0\]and solve for k.

Answer 25

Do you know what to do now?

Answer 26

I'll be right back, while you're working it out.

Answer 27

Okay. So then,

1(7-k) + 2(3-2k) + 3(5-3k) = 0

7 - k + 6 - 4k + 15 - 9k = 0

28 - 14k = 0

-14k = -28

k = 2

Answer 28

Eureka! So now can you tell me what are vectors v a and w?

Answer 29

v = (2, 4, 6) and w = (5, -1, -1)

Answer 30

So, (2, 4, 6) + (5, -1, -1) = (7, 3 ,5)

Answer 31

There you go! You got it dear Watson! Haha!

Answer 32

So do I deserve a medal? lol

Answer 33

Oh my god! You're incredible :)

Answer 34

Thank you even though I had to take some time to think about how to explain it to you.

Answer 35

Doesn't it feel great when you get the right answer? I LOVE MATH!

Answer 36

I think I gave you a medal, if I did that correctly. But, it doesn't matter that it took you time to think about it. Math isn't easy; its a challenging subject! You got me there. That is whats important!

Answer 37

Thank you and yes you did give a medal. I'm glad I was able to help you because believe me, I wouldn't have been able to sleep otherwise. lol Now I can rest easy.

Answer 38

Aw, how funny! Well I'm happy about that. I really appreciate your help. If you don't mind, I have other questions. But if you'd like to help, can it wait until tomorrow? I have an early class and have to get sleep myself!

Answer 39

Definitely! Tomorrow works better for me too because I have a couple of other people who are messaging me for help, and then I have to log out. Take care and have a great night!

Answer 40

I don't know what times I'll be on tomorrow but if you see that I'm on just try to grab my attention.

Answer 41

Okay. Thank you. I will try to message you/post a question in the afternoon. THANKS AGAIN! @calculusfunctions