Vector Magnitude & Direction Calculator
Calculate vector magnitude, direction angle, unit vectors, and perform vector addition and dot products in 2D and 3D. With step-by-step working.
Vectors Guide
Magnitude and Direction
A vector has both magnitude (size) and direction, unlike a scalar which has magnitude only. The magnitude of a 2D vector (x, y) = √(x² + y²). For vector (3, 4): magnitude = √(9+16) = √25 = 5. The direction angle θ from the positive x-axis: θ = arctan(y/x). For (3, 4): θ = arctan(4/3) = 53.1°. For 3D vector (x, y, z): magnitude = √(x² + y² + z²). The unit vector (a vector of magnitude 1 in the same direction) = original vector divided by its magnitude: (3/5, 4/5) = (0.6, 0.8) for our example.
Vector Addition
Vectors add component by component: (a, b) + (c, d) = (a+c, b+d). Geometrically, this is the tip-to-tail rule — place the second vector at the tip of the first; the resultant goes from the tail of the first to the tip of the second. Example: velocity vector (30, 0) m/s (boat moving east) plus current vector (0, 5) m/s (current moving north) = resultant (30, 5) m/s. Magnitude: √(900+25) = 30.4 m/s. Direction: arctan(5/30) = 9.5° north of east.
The Dot Product
The dot product of two vectors a·b = a₁b₁ + a₂b₂ (2D) or a₁b₁ + a₂b₂ + a₃b₃ (3D). It also equals |a||b|cos(θ), where θ is the angle between them. This property is used to find the angle between vectors. If the dot product = 0, the vectors are perpendicular (cos 90° = 0). Applications: calculating work done (W = F·d, force dotted with displacement), finding the angle of incidence in optics and collision physics, projecting one vector onto another in navigation.
Vectors in Physics and Engineering
Vectors appear throughout physics and engineering. Forces are vectors — adding the weight vector (downward) and the normal reaction vector (perpendicular to surface) gives the net force. Velocity and acceleration are vectors. Electric and magnetic fields are vector fields. Structural engineering: the equilibrium of forces (all vectors summing to zero) determines whether a structure is stable. Navigation: velocity vectors (speed and direction) combined with wind or current vectors give the actual
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