Dot Products and Projections HELP PLZ (1 Viewer)

csi

Member
Joined
Nov 10, 2019
Messages
94
Gender
Undisclosed
HSC
2021
Hi guys,

I'm stuck on the following questions, please give me a hand.
Note: all the letters (vectors) below are meant to have a tilde beneath it, it's just me not knowing how to type it out :(
1. If a·b=3, a·c=4, b·c=-2 and |b| =1 then evaluate (a-b)·(2b+3c)
ANS: 22

2. Find the possible values of k if the projection of ki+4j onto 12i-5j has length 140/12
ANS: 40/3

3. Points P and A are on the number plane. The vector is OA is (3,1). Point B is chosen so that the area of triangle PAB is 10u^2 and |OB| is 4 root 5. Find all possible vectors for OP.

Thanks!!!
 
Last edited:

CM_Tutor

Moderator
Moderator
Joined
Mar 11, 2004
Messages
2,644
Gender
Male
HSC
N/A
Note: all the letters (vectors) below are meant to have a tilde beneath it, it's just me not knowing how to type it out :(
I've been wondering the same as the methods I am aware of / can find by searching don't work here... so I have posted a question on it.

https://community.boredofstudies.org/threads/vectors-and-tex.394068/

I haven't had any answers so far, however. @Trebla has just replied that packages are needed for this :( and has also suggested the bold notation that I describe below.

While we wait, we can use other conventions. In the following, replace each hyphen (-) with a slash(\):

Display vectors as bold:



Display vectors using the arrow above notation using the command \overrightarrow:



 
Last edited:
  • Like
Reactions: csi

CM_Tutor

Moderator
Moderator
Joined
Mar 11, 2004
Messages
2,644
Gender
Male
HSC
N/A
I'll start... hopefully others will jump in too :)

Question 1



then evaluate



Observations:
  • Expansions of expressions like the one we are given are just like binomial expansions
  • Dot products are commutative, and so
  • The magnitude of a vector, , is related to the dot product by
So, the answer is:

 
  • Like
Reactions: csi

csi

Member
Joined
Nov 10, 2019
Messages
94
Gender
Undisclosed
HSC
2021
I'll start... hopefully others will jump in too :)

Question 1



then evaluate



Observations:
  • Expansions of expressions like the one we are given are just like binomial expansions
  • Dot products are commutative, and so
  • The magnitude of a vector, , is related to the dot product by
So, the answer is:

Thanks!
 

CM_Tutor

Moderator
Moderator
Joined
Mar 11, 2004
Messages
2,644
Gender
Male
HSC
N/A
Question 2



---












I'm not sure if I have made an arithmetic mistake or not, but I am surprised that there is only supposedly one answer for and I am also surprised that the length would be given as a fraction with the numerator and denominator having a common factor. @csi, would you please check the question?
 

CM_Tutor

Moderator
Moderator
Joined
Mar 11, 2004
Messages
2,644
Gender
Male
HSC
N/A
I'm not sure if I have made an arithmetic mistake or not, but I am surprised that there is only supposedly one answer for and I am also surprised that the length would be given as a fraction with the numerator and denominator having a common factor. @csi, would you please check the question?
For example, if the distance was rather than , we would have



So, I strongly suspect that the distance should be and that there are two answers, and
 
  • Like
Reactions: csi

Qeru

Well-Known Member
Joined
Dec 30, 2020
Messages
368
Gender
Male
HSC
2021
Question 2



---












I'm not sure if I have made an arithmetic mistake or not, but I am surprised that there is only supposedly one answer for and I am also surprised that the length would be given as a fraction with the numerator and denominator having a common factor. @csi, would you please check the question?
How long did that take you to type out
 

CM_Tutor

Moderator
Moderator
Joined
Mar 11, 2004
Messages
2,644
Gender
Male
HSC
N/A
How long did that take you to type out
Ummm... a while. Would have been faster if I wasn't trying to figure out why the numbers were getting messy. And if I wasn't an episode of Dr Who in the background at the same time. :)
 

csi

Member
Joined
Nov 10, 2019
Messages
94
Gender
Undisclosed
HSC
2021
For example, if the distance was rather than , we would have



So, I strongly suspect that the distance should be and that there are two answers, and
Yep, you're right. The length is meant to be 140/13 instead of 140/12. That's my bad, sorry about that!!
 

CM_Tutor

Moderator
Moderator
Joined
Mar 11, 2004
Messages
2,644
Gender
Male
HSC
N/A
Yep, you're right. The length is meant to be 140/13 instead of 140/12. That's my bad, sorry about that!!
Ok, that's good. Does the source give only one solution, or two?
 

csi

Member
Joined
Nov 10, 2019
Messages
94
Gender
Undisclosed
HSC
2021
Ok, that's good. Does the source give only one solution, or two?
Just 40/3, it does say all possible solutions though. I wouldn’t be surprised if the the answers were wrong, cause it happens all the time.
 

CM_Tutor

Moderator
Moderator
Joined
Mar 11, 2004
Messages
2,644
Gender
Male
HSC
N/A
Just 40/3, it does say all possible solutions though. I wouldn’t be surprised if the the answers were wrong, cause it happens all the time.
Ok, well, I think that's wrong. I think that there are two solutions. If you consider what a projection of one vector onto another is, there can and often will be two vectors that can be projected to give projections of the same length.

Is this from a textbook? If so, which? What's the reference (exercise, page, etc)?
 

Qeru

Well-Known Member
Joined
Dec 30, 2020
Messages
368
Gender
Male
HSC
2021
For example, if the distance was rather than , we would have



So, I strongly suspect that the distance should be and that there are two answers, and
Wow thats really cool I just realised you did 140/13 since the RHS has 13^4 in the denominator, will that always output a 'nice' answer for k?
 

CM_Tutor

Moderator
Moderator
Joined
Mar 11, 2004
Messages
2,644
Gender
Male
HSC
N/A
Wow thats really cool I just realised you did 140/13 since the RHS has 13^4 in the denominator
I played a hunch that the 140 / 12 was a typo and, as you said, the 13's in the answer made 140 / 13 a likely guess. :)
 
  • Like
Reactions: csi

csi

Member
Joined
Nov 10, 2019
Messages
94
Gender
Undisclosed
HSC
2021
Ok, well, I think that's wrong. I think that there are two solutions. If you consider what a projection of one vector onto another is, there can and often will be two vectors that can be projected to give projections of the same length.

Is this from a textbook? If so, which? What's the reference (exercise, page, etc)?
Nope, it’s not from a textbook.
 

CM_Tutor

Moderator
Moderator
Joined
Mar 11, 2004
Messages
2,644
Gender
Male
HSC
N/A
Wow thats really cool I just realised you did 140/13 since the RHS has 13^4 in the denominator, will that always output a 'nice' answer for k?
Sorry, I just realised that I missed that there was a question here... will it "always" give a nicer ? I couldn't say always, but you are right that seeing the terms with did make me think that a in the denominator was likely to help get a nicer answer, and the fact that 13 is a one-key typo from 12 made the possibility seem much more likely. :)

PS: @csi's question 3 above is addressed in another thread and I think the question has infinite answers, you might like to have a look and see if my reasoning is solid... you could even try to figure out all possible positions for ...
 

Users Who Are Viewing This Thread (Users: 0, Guests: 1)

Top