How do you find the resultant of a non-coplanar concurrent (space) force system?

In one line

Write each force as a vector using unit vectors along its line, add the x, y and z components, and get the resultant and its direction cosines.

Explained simply

A hot-air balloon tied down by three ropes to points on the ground: all ropes meet at the basket ring but point in different 3D directions. To find the net pull on the ring, you need components in three directions instead of two.

Technically, each force \( \vec{F} = F\,\hat{e}_{AB} \), where \( \hat{e}_{AB} \) is the unit vector from the point of application A towards B along the line. Then \( \vec{R} = \Sigma \vec{F} \).

How it works

1. Write coordinates of the common point A and of a second point B on each force's line.
2. Position vector \( \vec{AB} = (x_B-x_A)\hat{i} + (y_B-y_A)\hat{j} + (z_B-z_A)\hat{k} \).
3. Unit vector \( \hat{e}_{AB} = \vec{AB}/|AB| \).
4. Force vector \( \vec{F} = F \hat{e}_{AB} \).
5. Add all: \( \vec{R} = (\Sigma F_x)\hat{i} + (\Sigma F_y)\hat{j} + (\Sigma F_z)\hat{k} \).
6. Magnitude and direction cosines: \( R = \sqrt{\Sigma F_x^2 + \Sigma F_y^2 + \Sigma F_z^2} \), \( \cos\theta_x = \Sigma F_x/R \), etc.

Key formula

\( \vec{F} = F \dfrac{\vec{AB}}{|AB|} \); \( R = \sqrt{(\Sigma F_x)^2 + (\Sigma F_y)^2 + (\Sigma F_z)^2} \); \( \theta_x = \cos^{-1}(\Sigma F_x / R) \), similarly \( \theta_y, \theta_z \), with \( \cos^2\theta_x + \cos^2\theta_y + \cos^2\theta_z = 1 \). Forces in N, coordinates in m.

Quick example

A force of 130 N acts from A(0, 0, 0) towards B(3, 4, 12) m. \( |AB| = \sqrt{9 + 16 + 144} = 13 \) m. \( \vec{F} = 130(3\hat{i} + 4\hat{j} + 12\hat{k})/13 = 30\hat{i} + 40\hat{j} + 120\hat{k} \) N. Its angle with the z-axis is \( \cos^{-1}(120/130) = 22.6^\circ \).

Common mistakes

• Taking the vector from B to A when the force acts from A towards B (reverses every component).
• Forgetting to divide by the length, i.e. using \( F\vec{AB} \) instead of \( F\hat{e}_{AB} \).
• Giving only R without the direction cosines or angles.

How MU asks it

Usually a 5 to 6-mark numerical: 'Three forces act at the origin/at point A along the lines shown; find the resultant in magnitude and direction'. Write coordinates, each unit vector, each force vector, the sum, R and the three angles. Neat vector working earns method marks even if the arithmetic slips.