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I tried to solve the given question in the following way. Where I am going wrong?

Q. Let $\vec A$ and $\vec B$ be the two vectors of magnitude $10$ unit each. If they are inclined to the $x$-axis at angles $30^\circ$ and $60^\circ$ respectively, find the resultant.

I tried to solve the question in this way $\rightarrow$
Let, the angle between A and B = $\theta$
$\therefore\ \theta=60^\circ-30^\circ=30^\circ$
$\therefore\ |\vec R| = \sqrt{A^2+B^2+2AB \cos\theta}$
= $\sqrt{10^2+10^2+2\cdot10\cdot10\cos30^\circ}$
= $\sqrt{100+100+200\,\sqrt3/2}$
= $\sqrt{100+100+100\,\sqrt3}$
= $\sqrt{200+100\,\sqrt3}$
= $\sqrt{100\,(2+\sqrt3)}$
= $10\,\sqrt{2+\sqrt3}$

Now, I don't know how to solve it further. The answer is $20 \cos 15^\circ$. Please help me to simplify it further.

$$\cos 15^\circ = \frac{1+\sqrt3}{2\sqrt2}.$$

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  • $\begingroup$ Welcome to Physics Stack Exchange! Please take a minute to read our guidelines for homework and exercise questions as well as check-my-work questions. We intend our questions to be potentially useful to a broader set of users than just the one asking, and we prefer conceptual questions over those just asking for a specific computation. $\endgroup$
    – Emilio Pisanty
    Commented May 3, 2021 at 10:39
  • $\begingroup$ The question phrasing must also state that the two vectors should be added together or have some other arithmetic operation done. Otherwise it makes no sense to ask for the resultant vector. $\endgroup$
    – Steeven
    Commented May 3, 2021 at 10:48
  • $\begingroup$ To give a hint, I would write out the two vectors with their coordinates. And then add together those vectors, coordinate by coordinate. And then you have the resultant vector. If you need the magnitude, then you can extract/calculate it from this resultant vector - if you need the angle, then same. $\endgroup$
    – Steeven
    Commented May 3, 2021 at 10:51

1 Answer 1

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Your answer and the given answer both give the magnitude of the resultant vector (use your calculator). In this case the two given vectors have the same magnitude. The resultant will be halfway between them (at 45 degrees from the reference axis or 15 degrees from each of the given vectors. In the direction of the resultant, each contributes: 10 cos(15), for a total of: 20 cos(15). In general, for adding two vectors, I would find their, x, y, (and,z) components. Add components for each direction, and then recombine these to get the magnitude and direction of the resultant.

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  • $\begingroup$ Thanks, @R.W.Bird for the help. $\endgroup$
    – Mritunjay Kumar
    Commented May 4, 2021 at 2:14

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