Solving p and q in a Vector Equation

[Xaif]

Hello, I have a quick question that I think seems stupid but I can't find an answer for it. It's about vectors in a mechanics module. Anyhow, this is the question:

Determine p and q:

p(4i + 3j) + q(12i + 5j) = 68i + 33j, where p and q are contants.

I can only think to do this by trial and error to get p = 3.5 and q = 4.5. Is there a proper method to work out something like this?

 

[mattmns]

Yes there is.

If the components (i and j in this case) of two vectors are equal, then the two vectors are equal. So, in your equation, you need the components on one side to be equal to the components on the other side. This should give you two linear equations, which I am going to assume you can solve.

 

[AKG]

$$p(4\mathbf{i} + 3\mathbf{j}) + q(12\mathbf{i} + 5\mathbf{j}) = 68\mathbf{i} + 33\mathbf{j}$$

$$(4p + 12q)\mathbf{i} + (3p + 5q)\mathbf{j} = 68\mathbf{i} + 33\mathbf{j}$$

$$4p + 12q = 68\ \mbox{ and }\ 3p + 5q = 33$$

$$\left (\begin{array}{cc}4 & 12\\ 3 & 5\end{array}\right )\left (\begin{array}{c}p\\ q\end{array}\right ) = \left (\begin{array}{c}68\\ 33\end{array}\right )$$

$$\left (\begin{array}{c}p\\ q\end{array}\right ) = \left (\begin{array}{cc}4 & 12\\ 3 & 5\end{array}\right )^{-1}\left (\begin{array}{c}68\\ 33\end{array}\right )$$

 

[Xaif]

Ah, thankyou. It was quite simple really