Composing a Resultant Force Vector from Multiple Vectors

By Steve McCaw

Solving for the resultant force created when multiple forces act on a body involves several steps. The steps include using the tools of math and trigonometry to work with force vectors. Using a systematic approach makes it easier to arrive at the correct answer.

With vector quantities like force, the direction of the vector is as important as the magnitude. A force of +50 Newtons (N) in the vertical direction is different from a force of –50 N in the vertical direction. Pay attention to the magnitude and the direction of every force given in a problem you’re trying to solve. Similarly, your answer must provide both the magnitude and the direction for the resultant force.

When working with force vectors, be sure to first set a coordinate system to provide a reference for direction. Assign the positive and negative directions for both the horizontal and the vertical axis of your coordinate system. Sometimes this is set for you in the question, with words like “use upward as the + vertical direction.” Also identify the axis you’ll use when setting the direction of any vector with a direction given in degrees (for example, a force of 1,100 N at an angle of 38 degrees). Typically, the right horizontal axis represents 0 degrees, and the angle of a vector is measured as positive in the counterclockwise direction.

On your coordinate system, sketch out each vector given in the question. Show the positive vectors pointing in the positive direction, the negative vectors pointing in the negative direction, and any vector given in degrees pointing in the general direction of the given angle. Beside each arrow, assign each a name and write in the magnitude and the direction of each force (for example F1 = 300 N at 20 degrees, F2 = –830 N vertical, F3 = 1,100 N at 38 degrees). This step is important because it gives you a visual image of each vector.

Next, resolve each vector into its components. Components of a vector are at 90 degrees to each other. These are typically called the horizontal and vertical components. If the force is indicated as purely horizontal or purely vertical, this step is already done for you. For each vector with a direction that’s given as an angle, sketch out a right triangle to graphically show the two components. The given vector is the hypotenuse (H) of the right triangle. Assign the given angle as Ө, and use Ө to identify the side opposite (O) and the side adjacent (A).

The next step is important: Using your reference system, make sure you identify which of the opposite and the adjacent sides is the horizontal and which is the vertical component of your vector. Name each of these components with the force name and the component name (for example, F1H, F1V, F2H, F2V, F3H, F3V). Be sure to correctly align the adjacent and the opposite sides to the reference system. If you don’t do this, even if you complete the next step correctly, your calculated resultant force in the final step will be wrong.

Next, use one of the trigonometric functions — sine, cosine, or tangent — to calculate the magnitude of the individual sides of each right triangle using the given force (the hypotenuse) and the angle Ө. Use the anagram SOH CAH TOA to identify the correct trig function needed for each component of each vector.

You can remember the three trig functions using the letters SOH CAH TOA, which is short for the first letter of the trig function and the first letter of the two sides defined by the function:

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When you calculate each component, make sure you identify both the magnitude and the direction (+ or –) of the force.

The net force in each direction is the sum of all the forces acting in that direction, or Net ForceDirection = ΣForceDirection. For the horizontal direction, use ΣFH = F1H + F2H + F3H, and for the vertical direction use ΣFV = F1V + F2V + F3V. In each direction, use this format: ΣF = (Force) + (Force) + (Force). When entering the force vectors into the equation, enter both the magnitude and the direction (+ or –) within the parentheses. Now complete the summing to calculate the net force in each direction.

The final steps involve calculating the magnitude and direction of the resultant force created by the combined effect of the net force acting in the vertical direction and the net force acting in the positive direction. A diagram will help here. Draw the vector arrow representing the net horizontal force in the correct direction, and draw the vertical force vector arrow pointing in the correct direction (+ or –) with the tail of the vertical vector starting at the tip (arrowhead) of the horizontal force vector. Correctly label each of these sides as horizontal and vertical, and write in your calculated magnitude and direction (+ or –) of each force. The resultant force you will calculate is the hypotenuse of the right triangle you’ve sketched.

To calculate the magnitude of the resultant force, enter the net horizontal and vertical forces into the Pythagorean theorem (a2 = b2 + c2), or with your labeled sketch:

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To calculate the direction of the resultant force, enter the net horizontal and vertical force values into the trig function arctan:

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Present the answer in this format: The resultant force has a magnitude of (resultant magnitude) Newtons at an angle of Ө degrees.