Vectors in three dimensions
Real structures are three-dimensional: cables run from walls to corners, and towers lean in two directions at once. Cartesian vectors handle these cases systematically.
Cartesian vectors
In three dimensions, a vector is written \(\mathbf{F} = F_x\mathbf{i} + F_y\mathbf{j} + F_z\mathbf{k}\), with \(\mathbf{k}\) the unit vector along z. Its magnitude is \(F = \sqrt{F_x^2 + F_y^2 + F_z^2}\). The vector (300, 200, -600) N has magnitude \(\sqrt{90{,}000 + 40{,}000 + 360{,}000} = 700\) N.
Direction cosines
The angles \(\alpha\), \(\beta\), and \(\gamma\) between a vector and the x, y, and z axes satisfy \(\cos\alpha = \frac{F_x}{F}\), and similarly for y and z. For (300, 200, -600) N these are 3/7, 2/7, and -6/7. The direction cosines always satisfy \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\).
Position vectors
A position vector r from point A to point B is found by subtracting coordinates: B minus A. From A(0, 0, 6) m to B(3, 2, 0) m, r = (3, 2, -6) m, with length the square root of (9 + 4 + 36) = 7 m. Position vectors describe where cables and members run.
Force along a line
A force acting along a cable has the cable's direction, so \(\mathbf{F} = F\frac{\mathbf{r}}{r}\). A 700 N tension in the cable from A to B above is \(700 \times \frac{(3, 2, -6)}{7} = (300, 200, -600)\) N. This 'unit vector times magnitude' step appears in almost every 3D problem.
The dot product
The dot product \(\mathbf{A} \cdot \mathbf{B} = A_xB_x + A_yB_y + A_zB_z = AB\cos\theta\) gives the angle between two vectors. It also gives the component of a force along a line: the projection of F onto unit vector u is \(\mathbf{F} \cdot \mathbf{u}\). Perpendicular vectors have a dot product of zero.
Finding an angle
For \(\mathbf{A} = (2, 1, 2)\) and \(\mathbf{B} = (1, 0, 0)\): \(\mathbf{A} \cdot \mathbf{B} = 2\), \(A = 3\), and \(B = 1\), so \(\cos\theta = \tfrac{2}{3}\) and \(\theta \approx 48.2°\). Engineers use this to find angles between members and cables from coordinates alone.
Adding 3D forces
Resultants in 3D add component by component, exactly as in 2D but with a third coordinate. Writing every force as a Cartesian vector first, then summing, avoids errors with angles in perspective drawings, which are easy to misread.
Worked example
A radio mast is held by a cable from its top at (0, 0, 12) m to a ground anchor at (4, -3, 0) m with 2.6 kN of tension acting on the mast. The cable vector from top to anchor is (4, -3, -12) m, of length 13 m, so the force on the mast is 2.6 x (4, -3, -12)/13 = (0.8, -0.6, -2.4) kN. Its downward component, 2.4 kN, adds to the mast's compression.
Key takeaways
3D vectors are written with i, j, and k components, and their magnitude is the square root of the sum of squares. Direction cosines are components divided by magnitude, and their squares sum to 1. Position vectors subtract coordinates. A force along a line equals its magnitude times the line's unit vector. The dot product gives angles between vectors and projections of forces onto lines, and is zero for perpendicular vectors. Add 3D forces component by component.
Flashcard
Unit vector along z
Check your understanding
What is the magnitude of the force (300, 200, -600) N?