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Find vector collinear to sum of other two vectors

  1. Aug 21, 2009 #1
    1. The problem statement, all variables and given/known data

    Find vector [itex]\overrightarrow{c}[/itex] collinear with the vector [itex]\overrightarrow{a}+\overrightarrow{b}[/itex], if [itex]\overrightarrow{a} \cdot \overrightarrow{b}=5[/itex] and [itex]\overrightarrow{c} \cdot \overrightarrow{b} = 18[/itex], [itex]|\overrightarrow{b}|=2[/itex]


    2. Relevant equations

    [tex]|\overrightarrow{a} \times \overrightarrow{b}| = |\overrightarrow{a}||\overrightarrow{b}|sin(\overrightarrow{a},\overrightarrow{b})[/tex]

    [tex]\mathbf{a}\times\mathbf{b}=\det \begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ \end{bmatrix}.[/tex]

    [tex]\overrightarrow{a} \cdot \overrightarrow{b} = |\overrightarrow{a}||\overrightarrow{b}|cos(\overrightarrow{a},\overrightarrow{b})[/tex]

    3. The attempt at a solution

    1z1qxeb.png

    Before I started I found some (I believe) error in the task.

    If I subtract the given equations like a.b-c.b=-13 and b(a-c)=-13

    |b||a-c|cos(b, a-c) = -13

    As we can see on the picture a-c0=-b

    |b||a-pc0|cos(b,a-pc0)=-13

    Edit: Ok, there isn't error I mixed c0 and c.

    Now, let me start.

    I approach

    c = p(a+b), because c and a+b are collinear, where p is real number.

    From the given conditions.

    |a||b|cos(a,b)=5

    |c||b|cos(c,b)=18

    |a|cos(a,b)=5/2

    |c|cos(c,b)=18/2

    c.(a+b)=|c||a+b|cos(c,a+b)=|c||c0|cos(c,c0)=|c|

    Here is where I am stuck...:confused:

    II approach

    c x (a+b) = 0

    c x a + c x b =0

    But as you can see I got no coordinates for the vectors, so this is 2nd fail...:mad:

    I am dealing with this task for 1 hour and seems like I can't find any way to solve it.

    The result in the textbook is c = 2(a+b)

    Thanks in advance.

    Regards.
     
    Last edited: Aug 21, 2009
  2. jcsd
  3. Aug 21, 2009 #2

    kuruman

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    Homework Helper
    Gold Member

    Use the simple fact that if two vectors are collinear, one is a scalar multiple of the other. Then you can write

    c = k (a + b)

    All you need to do is find k. Do you see how to do that in view of the given information?
     
  4. Aug 21, 2009 #3

    kuruman

    User Avatar
    Homework Helper
    Gold Member

    After reading your attempt more carefully, I saw that my hint will not really help you because you've been there. What I call k you call p. So here comes the next hint, what do you get when you dot both sides with b?
     
    Last edited: Aug 21, 2009
  5. Aug 21, 2009 #4
    Aaah... I see now.

    c.b = (ka+kb).b

    18=k(a.b) +k(b.b)

    18=5k+4k

    9k=18

    k=2

    Thanks a mill. times, I didn't spot that thing (I thought it was something more complicated).

    Regards.
     
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