What topic is this question?

I did a level maths be it a while ago but I don't remember perpendicularity of 1) vectors giving more infomation in order to determine something about the vectors it is perpendicular to and 2) I cant think of any solution myself 3) nor do I know where to look to find infomation I need in order to work it out.

This should be covered (if not completely, to the point where you can understand what we're saying) in the vectors chapters from your A-levels math book.

Right I mean I don't have that anymore, hence why I ask, what is the *topic* e.g. if this were differentiation a sort of subset or topic within that may be differentials of functions such as sin or ln

What topic within the broad scope of "vectors" covers how you can gain information from the angle of one vector to the other

I cannot simply read through the whole internet on vectors to find this although I'm trying I feel I may miss it if I dont have a great memory of previous knowledge

Right I mean I don't have that anymore, hence why I ask, what is the *topic* e.g. if this were differentiation a sort of subset or topic within that may be differentials of functions such as sin or ln

What topic within the broad scope of "vectors" covers how you can gain information from the angle of one vector to the other

I cannot simply read through the whole internet on vectors to find this although I'm trying I feel I may miss it if I dont have a great memory of previous knowledge

It would be helpful to know your background / where the question came from. Most people asking for help here are gcse/a level hence you have a reasonable idea of how to pitch the help. Forum rules are that solutions are not done. You don't have great previous knowledge or seemingly the books?

Googling
Show vectors are perpendicular
Has the dot product in just about all of the links. You do not need to read the whole of the vector internet, maybe just the top link or two. Similarly, if you Google the question, you'll find it has been discussed in the past. Again, it does not require much effort. The "proof" (or counter example) is about 3 relatively simple lines. If you find it difficult, it suggests there is extra material you should brush up on first. It really depends on what your goals are, but it would be appreciated it you could answer the first questions.

What I have got thus far:

If u has to be perpendicular to both v*w and also v + 2w then v*w and v + 2w must be parrallel in order for the propositon? [is that the name for this?] to be true.

In order to be parrallel v and w must have the same reference angle angle or gradient. [should I just do the reference angle? tan^-1 17/7 it should end up being for v + 2w? its just gradient seems the simpler solution]

And so I will try and give a counter example as v*w [is that what v*w means?] always having the same gradient as v + 2w seems unlikely [would I write that in a proof?]

let v = <3 , 7> and w = <2 , 5>

v + 2w = (xv + 2xw) , (yv + 2yw) = (3 + 2*2) , (7 + 5*2) = <7 , 17> , therefore the gradient of v + 2w [forgotten the way to express this algebreicly m something or other?] = 17/7

v * w = [the dot product is a scalar quantity?]


Edit: ohhhh does v and w = v + w?

Can you pls give me an idea of your background/level and where the question comes from?
If so, I'll happily give some feedback.

This is a past paper question at uni year 1 I believe its supposed to be a easy starting question/refresher -> I studied at a level 2 years ago but I have not used my knowledge so this is difficult for me.

Can you tell me if what I say in the square brackets i.e. [ ] and in the edit is correct? Where have I gone wrong?

This is a past paper question at uni year 1 I believe its supposed to be a easy starting question/refresher -> I studied at a level 2 years ago but I have not used my knowledge so this is difficult for me.

Can you tell me if what I say in the square brackets i.e. [ ] and in the edit is correct? Where have I gone wrong?

So my issue in the question is 2) that I have 0 bloody idea how perpendicularity plays into the problem hence I have no idea what u really does in the context of the question.

Can I suggest you start again? Hopefully you'll understand more.

Two vectors u and v are perpendicular. How do you show this, and can you give me a simple example of u and v in two dimensions.

As both v and w have gradients meaning they are perpendicular then v and w must be multiples of each other therefore

Ohhh I've been misreading the question this whole time, I read v and w as a single vector but it is asking "when if v and w are both perpendicular to u then show that v + 2w must be perpendicular, if this proposition is false, give a counter example."

This seems harder, I believe I need to show that v and 2w logically must always have the same gradient as either v or w alone would have.

Thank you.

Here is my attempt:

The basic "plan of attack": https://imgur.com/a/V5Z5OrA

The write up:

As both v and w have gradients meaning they are perpendicular then v and w must be multiples of each other therefore v + 2w is simple a multiple of both either v or w hence it has the same gradient and therefore as the gradient of both v and w multiplied by the recipricol of the gradient of u is -1 and so are perpendicular the product of the gradient of v + 2w and the recipricol of the gradient of u must also be -1 [ok I suck at wording things]
[Is there a way of doing this algebreicly?]

[[e.g. to check

v = <2 , 5> , w = < 4, 10> , u = < 5 , 2 >

v * 1/[gradient of...?]u = -1 and w * 1/u = -1?

(v + 2w) * 1/u must also be -1?

v + 2w = <10 , 25>

Yeah I now see, how does one get the recipricol of u if that is indeed what I need?

]]

Why not re-read the previous post and start where I suggest.
You're far from the mark for a relatively simple question.

[So]

This is essentially unanswerable without knowing more about what you are actually studying.

OK, I think you mean by that, there are proofs but you assume they want a specific proof?

Can you tell me what operation we can use to see if two (non-zero) vectors are perpendicular to each other?

If not I suggest you look at perpendicular vectors which should mention a specific test to use.

Also, more broadly, for mathematics at university, definitions play a very important role especially if you are proving something is a certain object where you need to satisfy the definition of the object.
You should see definitions scattered in the the class notes distributed by your lecturer and mathematical texts such as here:


(screenshot form pdf of book "Measure, Integration and Real Analysis" by Sheldon Axler, taken 18/10/2020 if interested).

If they expect an A-level understanding of what a vector is, you can't use gradients - they don't work beyond 2 dimensions. But maybe they only expect GCSE/AS knowledge?

This is going to be my last response to you until you answer the questions about what you're actually studying and so on.