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    So basically I have been self-assessing Core 4 (AQA) and the topic that I have problems with, mainly due to my text books not providing enough information on each method that are essential for the topic, is the one about vectors. I know must of it but there are three questions that I have been having problems answering...

    1. L1 = \begin{bmatrix} -3\\ -2 \\ -3 \end{bmatrix} + s\begin{bmatrix} 1\\ 1 \\ 0 \end{bmatrix} L2 = \begin{bmatrix} 4\\ 3 \\ 5 \end{bmatrix} + t\begin{bmatrix} 3\\ 1 \\ 2 \end{bmatrix}

    the point A = \begin{bmatrix} 1\\ 1 \\ 1 \end{bmatrix}. Find the co-ordinates of the foot of the perpendicular from A to L1.

    2. Given that the two vectors, \begin{bmatrix} 1+s\\ 2 \\ s \end{bmatrix}, \begin{bmatrix} -1\\ 1-s \\ 2 \end{bmatrix} are perpendicular, find s.

    3. Show that the shortest distance between the two lines r1 = \begin{bmatrix} 1\\ 2 \\ 0 \end{bmatrix} + s\begin{bmatrix} -1\\ 0 \\ 3 \end{bmatrix} and r2 = t\begin{bmatrix} 0\\ 4 \\ 0 \end{bmatrix} is 3\sqrt 2

    I will try to rep anyone who helps to explain to me how to go about these three questions. I think once I'm sorted with these three questions, I should be fine with vectors, ultimately.

    Thank you for you patience :p:
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    1) By definition the perpendicular from A to L1 and L1 are orthogonal. What can you deduce from this (in terms of dot products)?

    2) See above

    3) If you think about it the line of shortest distance between two lines is orthogonal to both lines.
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    I'm not too sure really, I have yet to come across the theory of dot products (if you could nicely explain what they are that would definitely help). if it is not too much to ask but could you walk me through those two questions.
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    (Original post by Summerdays)
    if it is not too much to ask but could you walk me through those two questions.
    That's not really how these forums work, I'm afraid - we're quite happy to give you hints, but providing worked solutions is in most cases against the rules.

    Do you know anything about dot products? You don't strictly need to, but it makes this question a lot easier.
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    Ah wait, you already said you know nothing about dot products. Ok...

    An alternative (but much slower) way is to take the point P(s) = (-3, -2, -3) + s(1, 1, 0) and the point A. Work out the square of the distance AP, and then minimise it (by differentiating or whatever). This tells you the minimum distance, and hence the perpendicular distance.
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    Oh, sorry, for my C4 course we did dot products at the same time as vectors. If you don't know about them I'd suggest you learn, since they make various perpendicularity questions a lot easier.

    Basically, the dot product of two vectors \vec{a} . \vec{b} = |a| |b| cos \theta where \theta is the angle between the two vectors. Hence if two vectors are perpendicular, \theta = 90^{\circ} and thus the dot product is equal to zero. This seems fine and dandy, but it's not much use in the form given here. However, an alternative way of computing the dot product of two vectors l_1 = \begin{pmatrix} a_1 \\ b_1 \\ c_1 \end{pmatrix} and l_2 = \begin{pmatrix} a_2 \\ b_2 \\ c_2\end{pmatrix} is given by  a_1 \times a_2 + b_1 \times b_2 + c_1 \times c_2

    Does that help a bit?
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    Oh yes, I know about this rule I just didn't realise it was called "dot product". Oh yes, that is quite silly of me; this makes question two a sinch. Thank you very much.

    EDIT: From your information, the answer for question 1 is (-2,-1, 3) and the answer for question two is s = -1?

    I'm still unsure about question three; how do I go about that?
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    (Original post by Summerdays)
    Oh yes, I know about this rule I just didn't realise it was called "dot product". Oh yes, that is quite silly of me; this makes question two a sinch. Thank you very much.

    EDIT: From your information, the answer for question 1 is (-2,-1, 3) and the answer for question two is s = -1?

    I'm still unsure about question three; how do I go about that?
    I have a different answer for question 1 i.e (0.5,1.5,-3) Are you saying your answer is (-2,-1,3) or the answer in the book is (-2,-1,3) ?

    I am rusty so may well be wrong.
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    That is what I deduced for question one, but I'm probably wrong because I'm not too sure about those types of questions myself. BTW what method did yo uuse to get your answer?
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    I can't be bothered to work this out so i'm just going to say the answer is 5
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    (Original post by Summerdays)
    I'm still unsure about question three; how do I go about that?
    I'm not sure myself, but on a quick googling, you might find this page helpful on solving your third question. :smile:

    (Original post by radiated yoghurt)
    I can't be bothered to work this out so i'm just going to say the answer is 5
    One of the most pointless and useless posts I've seen in a while.
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    (Original post by Summerdays)
    3. Show that the shortest distance between the two lines r1 = \begin{bmatrix} 1\\ 2 \\ 0 \end{bmatrix} + s\begin{bmatrix} -1\\ 0 \\ 3 \end{bmatrix} and r2 = t\begin{bmatrix} 0\\ 4 \\ 0 \end{bmatrix} is 3\sqrt 2
    Generally finding the distance between two lines would be a furter maths problem rather than C4.

    In this case however it's a 2 dimensional problem since the first line lies in the plane y=2 and the second line is the y axis.

    By the way you will find that when s=0 and t=1/2 the distance is 1.

    And the shortest distance is \frac{3}{\sqrt{10}}
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    (Original post by rnd)
    Generally finding the distance between two lines would be a furter maths problem rather than C4.

    In this case however it's a 2 dimensional problem since the first line lies in the plane y=2 and the second line is the y axis.

    By the way you will find that when s=0 and t=1/2 the distance is 1.

    And the shortest distance is \frac{3}{\sqrt{10}}
    I actually got my shortest distance to be equal to \frac{3\sqrt{10}}{10}}, but that is contrary to the actual question
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    As a matter of interest where did you get these questions. The last question in particular seems rather difficult for C4, which makes me wonder the source of the questions.

    If you are working on AQA C4 then you might find the past papers AND answers on the AQA website useful.
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    There were Review sheet questions that my college particularly like to handout as homework. I do agree that they are particularly hard for C4. Those questions seemed more like FP2 (MEI)
 
 
 
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