Find vector collinear to sum of other two vectors

[Дьявол]

Homework Statement

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

Homework Equations

|\overrightarrow{a} \times \overrightarrow{b}| = |\overrightarrow{a}||\overrightarrow{b}|sin(\overrightarrow{a},\overrightarrow{b})

\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}.

\overrightarrow{a} \cdot \overrightarrow{b} = |\overrightarrow{a}||\overrightarrow{b}|cos(\overrightarrow{a},\overrightarrow{b})

The Attempt at a Solution

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...

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...

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.

 

[kuruman]

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?

 

[kuruman]

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?

 

[Дьявол]

After reading your attempt more carefully, I saw that my hint will not really help you because you've 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?

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.