Calculate p and q for Multiplying Vectors

[rocomath]

Three vectors have magnitudes a=3.00m (on the positive x-axis), b=4.00m (makes an angle of 30 degrees with the positive x-axis, counter-clockwise), and c=10.0m (makes an angle of 60 degrees with the negative x-axis, clockwise). If $$\overrightarrow{c}=p\overrightarrow{a}+q\overrightarrow{b}$$ what are the values of p and q?

I know that the dot product is

$$\overrightarrow{c}=\overrightarrow{a} \cdot \overrightarrow{b}$$

So...

$$\overrightarrow{a} \cdot \overrightarrow{b}=p\overrightarrow{a}+q\overrightarrow{b}$$

I'm not really sure what my next step should be.

 

[berkeman]

Hint: Draw the 3 vectors, with the head of a touching the tail of b, and the tail of c touching the tail of a. Do you see that you could ratio the sizes of a and b (remember that p & q can be negative numbers if needed), to make the head of the sum of pa+qb meet the head of c?

 

[neutrino]

$$\overrightarrow{c}=\overrightarrow{a} \cdot \overrightarrow{b}$$$$\overrightarrow{a} \cdot \overrightarrow{b}=p\overrightarrow{a}+q\overrightarrow{b}$$

Those are not even mathematically valid equations! In the first one, you have a vector on the left and a scalar on the right. (Remember, the dot product of two vector results in a scalar.) In the second, it's the other way around.

The simplest way, if you ask me, is to equate components and solve the resulting set of linear equations.

 

[D H]

$$\overrightarrow{c}=p\overrightarrow{a}+q\overrightarrow{b}$$
I'm not really sure what my next step should be.

As neutrino noted, your approach is not valid. You are on the right track, however. Taking the inner product with $\vec a$ and $\vec b$ yields a pair of linear equations in $p$ and $q$:

$$\begin{aligned}<br /> \vec a \cdot \vec c &= \vec a \cdot \vec a \, p + \vec a \cdot \vec b \, q \\<br /> \vec b \cdot \vec c &= \vec b \cdot \vec a \, p + \vec b \cdot \vec b \, q<br /> \end{aligned}$$

You have the requisite information needed to determine all of the inner products in the above equations. Solve the system of equations for $p$ and $q$, and voila, you have the answer.

 

[rocomath]

Those are not even mathematically valid equations!

$$\begin{aligned}<br /> \vec a \cdot \vec c &= \vec a \cdot \vec a \, p + \vec a \cdot \vec b \, q \\<br /> \vec b \cdot \vec c &= \vec b \cdot \vec a \, p + \vec b \cdot \vec b \, q<br /> \end{aligned}$$

Can't believe I made such a dumb mistake. Alright, I will attempt this problem again with your suggestions, thanks!