Resultant Vector Formula with Worked Examples

Key Points

• The resultant vector formula is $R = \sqrt{A^2+B^2+2AB\cos\theta}$, where $\theta$ is the angle between the two vectors.

• Magnitudes cannot simply be added, because two vectors only combine arithmetically when they point the same way.

• The angle $\theta$ must be measured between the vectors drawn tail to tail, not between a vector and an axis.

• The direction of the resultant is $\tan\alpha = \frac{B\sin\theta}{A+B\cos\theta}$, measured from the vector of magnitude $A$.

• At $\theta = 90$ degrees the cosine term vanishes and the formula reduces to Pythagoras, $R = \sqrt{A^2+B^2}$.

What Is the Resultant Vector Formula?

The resultant vector formula gives the magnitude of the single vector equivalent to two others acting together:

$$R = \sqrt{A^2 + B^2 + 2AB\cos\theta}$$

This article covers the magnitude and direction formulas. For the operation itself and the triangle and parallelogram laws that justify it, see addition of vectors.

Why Magnitudes Do Not Simply Add

Two forces of 3 N and 4 N do not always give 7 N. They give 7 N only when they pull in exactly the same direction, and 1 N when they pull in exactly opposite directions.

Everything in between depends on the angle, and that is what the $2AB\cos\theta$ term carries. At $\theta = 0$ the cosine is 1 and the expression becomes $(A+B)^2$ under the root, giving $A+B$. At $\theta = 180$ degrees the cosine is $-1$ and it becomes $(A-B)^2$, giving $\lvert A-B \rvert$.

So the resultant always lies between $\lvert A-B \rvert$ and $A+B$. Any answer outside that range is wrong before the arithmetic is checked.

Variable Key

Examples of the Resultant Vector Formula

Example 1

Two vectors of magnitude 3 and 4 act at 90 degrees. Find the resultant.

$$R = \sqrt{9+16+2(3)(4)\cos 90^\circ} = \sqrt{25+0} = 5$$

Final answer: 5. With $\cos 90^\circ = 0$ the formula reduces to Pythagoras.

Example 2

Two forces of 5 N and 3 N act at 60 degrees. Find the resultant by adding the magnitudes.

Adding gives $5 + 3 = 8$ N, which looks like a reasonable total.

But 8 N is the answer only if both forces pull in exactly the same direction. These act 60 degrees apart, so part of each force pulls across the other rather than alongside it, and the combined pull must be less than 8.

Using the formula with $\cos 60^\circ = 0.5$:

$$R = \sqrt{25+9+2(5)(3)(0.5)} = \sqrt{25+9+15} = \sqrt{49} = 7$$

Final answer: 7 N. The sum 8 N is the upper bound, reached only at zero degrees.

Example 3

Two equal vectors of magnitude 6 act at 120 degrees. Find the resultant.

$$R = \sqrt{36+36+2(36)(-0.5)} = \sqrt{72-36} = \sqrt{36} = 6$$

Final answer: 6

Two equal vectors at 120 degrees produce a resultant of the same magnitude as each one, which is why three such vectors at 120 degrees apart cancel completely.

Example 4

Find the direction of the resultant in Example 2, measured from the 5 N force.

$$\tan\alpha = \frac{3\sin 60^\circ}{5+3\cos 60^\circ} = \frac{2.598}{6.5} \approx 0.3997$$ $$\alpha \approx 21.79^\circ$$

Final answer: about $21.79^\circ$ from the 5 N force.

The resultant leans towards the larger vector, which is the expected behaviour and a quick check on any direction answer.

Example 5

Two vectors of magnitude 8 and 5 act at 180 degrees. Find the resultant.

$$R = \sqrt{64+25+2(40)(-1)} = \sqrt{89-80} = \sqrt{9} = 3$$

Final answer: 3, which is $\lvert 8-5 \rvert$, the lower bound of the range.

Two Routes to the Same Resultant

The component method scales better. Adding five vectors by the magnitude formula means four separate applications, whereas resolving each into components lets all the $x$ parts and all the $y$ parts be summed in one pass.

Resultant Vector, Where Things Go Sideways

Mistake 1: Adding the magnitudes

Where it slips in: The vectors are given as numbers, and numbers invite addition.

Don't do this: $R = A + B$

The correct way: $R = \sqrt{A^2+B^2+2AB\cos\theta}$. Plain addition is the special case $\theta = 0$, and nothing else.

Students who write down the two bounds, $\lvert A-B \rvert$ and $A+B$, before computing anything can see immediately whether their answer is plausible. It costs one line and rules out the whole class of magnitude-addition errors.

Mistake 2: Using the wrong angle

Where it slips in: A diagram may show a vector's angle to the horizontal, which is not the angle between the two vectors.

Don't do this: Substitute a vector's bearing for $\theta$.

The correct way: Redraw the vectors tail to tail and measure the angle between them. If each vector's angle to the axis is given, $\theta$ is the difference of those two angles.

Mistake 3: Dropping the cosine term

Where it slips in: The first two terms look like Pythagoras, which is the familiar pattern.

Don't do this: $R = \sqrt{A^2+B^2}$ for any angle.

The correct way: Keep $2AB\cos\theta$ unless the angle is exactly 90 degrees. Omitting it understates the resultant for acute angles and overstates it for obtuse ones.

Key Takeaways

• The resultant vector formula is $R = \sqrt{A^2+B^2+2AB\cos\theta}$.

• Magnitudes add only when the vectors are parallel, which is the $\theta = 0$ case.

• The resultant always lies between $\lvert A-B\rvert$ and $A+B$, which makes a plausibility check instant.

• The angle must be measured between the vectors tail to tail, not from an axis.

• To work through vectors with a teacher, explore a trigonometry tutor or math tuition.

Practice These Three Before Moving On

Find the resultant of 7 and 24 at 90 degrees. Then find the resultant of 10 and 10 at 60 degrees, and state the bounds for two vectors of 9 and 4 before computing anything.

Want a live Bhanzu trainer to walk through more vector problems? Book a free demo class.

Read More

• Product of Vectors

• Collinear Vectors

• Multiplication of Vectors

• Angle Between Vectors

• Sin Cos Formula