Direction Cosines in 3D Space (\(l, m, n\))

Direction Angles \(\alpha, \beta, \gamma\) & Fundamental Identity \(l^2 + m^2 + n^2 = 1\)

Coordinates of Point P(x, y, z)

P(3.0, 4.0, 5.0)

x (along X-axis): 3.0

y (along Y-axis): 4.0

z (along Z-axis): 5.0

\(\alpha\) (with +X) 64.9° 1.13 rad

\(\beta\) (with +Y) 55.6° 0.97 rad

\(\gamma\) (with +Z) 45.0° 0.79 rad

r = |OP| = 7.07

\(l = \cos\alpha = \frac{x}{r}\) 0.424

\(m = \cos\beta = \frac{y}{r}\) 0.566

\(n = \cos\gamma = \frac{z}{r}\) 0.707

Identity: \(l^2 + m^2 + n^2 = 1\) VERIFIED

(0.424)² + (0.566)² + (0.707)² = 1.000

Angle Arcs (\(\alpha, \beta, \gamma\)) 3D Coordinate Projection Box Perpendicular Projections to Axes Ground Grid

X-Axis

Y-Axis

Z-Axis

Vector \(\vec{OP}\)

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