Determining Vector Resultant Direction with Sine Rule

Resultant Direction Needs a New Angle

The cosine rule gives the resultant magnitude, but it does not fully give the direction. In vector problems, an answer such as $15.6\text{ N}$ is incomplete if the force direction is not stated.

For two vectors, the resultant direction can be found using the sine rule. This rule works on the resultant triangle formed by the two vectors and their resultant.

\frac{\sin\phi}{B}=\frac{\sin\theta}{R}

Here, $\theta$ is the included angle between $\vec{A}$ and $\vec{B}$ , $R$ is the resultant magnitude, and $\phi$ is the angle between $\vec{A}$ and resultant $\vec{R}$ .

The sine rule is usually used after the resultant magnitude is known.

Reading the Angle Being Asked For

Before calculating, decide the reference direction. If the question asks for the resultant direction relative to $\vec{A}$ , the required angle is from $\vec{A}$ toward $\vec{R}$ .

If the question asks for direction relative to $\vec{B}$ , the angle is different. Do not swap angles without checking the vector triangle.

A common relationship is:

\sin\phi=\frac{B\sin\theta}{R}

This formula finds the resultant angle relative to vector $\vec{A}$ . Because the side opposite angle $\phi$ has magnitude $B$ , $B$ appears in the numerator.

Resultant Force Direction

Two forces act at one point. $A=10\text{ N}$ , $B=8\text{ N}$ , and the included angle is $60^\circ$ . From the cosine rule, the resultant magnitude is $R\approx15.6\text{ N}$ .

Now find angle $\phi$ between $\vec{A}$ and $\vec{R}$ .

\begin{aligned} \sin\phi&=\frac{B\sin\theta}{R} \\ &=\frac{8\sin60^\circ}{15.6} \\ &=\frac{8(0.866)}{15.6} \\ &\approx0.444 \end{aligned}

Therefore:

\phi=\sin^{-1}(0.444)\approx26.4^\circ

So the resultant force is about $15.6\text{ N}$ at $26.4^\circ$ from $\vec{A}$ toward $\vec{B}$ .

Checking with Components

The same direction can be checked using components. If $\vec{A}$ is placed on the positive $x$ axis, then:

R_x=A+B\cos\theta

For the same example:

\tan\phi=\frac{8\sin60^\circ}{10+8\cos60^\circ}=\frac{6.928}{14}\approx0.495

This gives $\phi\approx26.4^\circ$ , matching the sine rule. This check is useful when you are unsure which angle is being found.

When the Sine Rule Is Less Practical

The sine rule is less practical when there are more than two vectors or when each vector direction is already given relative to coordinate axes. In those cases, the component method is usually safer because all directions are organized along the $x$ and $y$ axes.

Use the sine rule when the problem is clear: two vectors, a known included angle, a known resultant magnitude, and a requested resultant direction relative to one of the vectors.