Vector Magnitude, Addition & Dot Product Calculator

Vector Guide

What's the key thing to understand about Vector Magnitude and Unit Vectors?

For vector a = (x, y): |a| = √(x² + y²). For 3D a = (x, y, z): |a| = √(x² + y² + z²). Unit vector â = a / |a| — a vector of magnitude 1 in the direction of a. Example: a = (3, 4). |a| = √(9+16) = 5. â = (3/5, 4/5) = (0.6, 0.8). Check: |â| = √(0.36 + 0.64) = 1 ✓. Standard basis vectors: i = (1,0,0), j = (0,1,0), k = (0,0,1). Any 3D vector: a = a_x i + a_y j + a_z k.

What do I need to know about Vector Addition and Subtraction?

Component-wise: (a_x + b_x, a_y + b_y). Geometrically: tip-to-tail addition (place vectors head to tail). Parallelogram law: resultant is the diagonal. Example: a=(3,4), b=(1,2). a+b = (4,6). |a+b| = √(16+36) = √52 = 7.21. Triangle of forces: three forces in equilibrium form a closed triangle. Resolution of forces: any force can be resolved into perpendicular components: F_horizontal = F cosθ, F_vertical = F sinθ. This is used constantly in mechanics problems.

What do I need to know about Dot Product?

a·b = a_x b_x + a_y b_y + a_z b_z = |a||b|cos θ. If a·b = 0: vectors are perpendicular (cos 90° = 0). If a·b > 0: angle < 90°. If a·b < 0: angle > 90°. Finding angle: θ = arccos(a·b / (|a||b|)). Example: a=(3,4), b=(1,2). a·b = 3+8 = 11. |a|=5, |b|=√5=2.236. cos θ = 11/(5×2.236) = 0.984. θ = arccos(0.984) = 10.3°. Work done: W = F·d = |F||d|cos θ. Only the component of force in the direction of motion does work.

What's the key thing to understand about Cross Product?

a × b = (a_y b_z − a_z b_y, a_z b_x − a_x b_z, a_x b_y − a_y b_x). |a × b| = |a||b|sin θ. Direction: perpendicular to both a and b (right-hand rule). If a × b = 0: vectors are parallel. Applications: torque τ = r × F. Angular momentum L = r × p. Area of parallelogram formed by two vectors = |a × b|. Normal to a surface: cross product of two vectors in the plane gives the surface normal. The cross product only exists for 3D vectors — no 2D equivalent.

What do I need to know about the Angle Between Two Vectors?

The dot product formula rearranges to give the angle between any two vectors directly: cos(θ) = (a·b) / (|a||b|). For a=(3,4) and b=(4,3): a·b = 3×4+4×3 = 24. |a|=5, |b|=5. cos(θ) = 24/25 = 0.96, so θ = cos⁻¹(0.96) ≈ 16.3°. This works in any number of dimensions, unlike the cross product (which only gives a vector in 3D). If a·b = 0, the vectors are perpendicular regardless of their magnitudes — a quick test used constantly in physics and graphics to check orthogonality without computing an actual angle. If a·b is negative, the angle between the vectors is greater than 90°. This is the formula behind calculating the angle between two forces, two edges of a 3D model, or two directions of motion.

What should I know about Scalar and Vector Projections?

Projecting vector a onto vector b answers 'how much of a points in the direction of b?' The scalar projection is (a·b)/|b|, and the vector projection scales b's unit vector by that amount: proj_b(a) = [(a·b)/|b|²] × b. Example: a=(3,4), b=(1,0). a·b=3, |b|=1, so the scalar projection is 3 — meaning a has a component of length 3 along the x-axis, which matches a's own x-component since b is the x-axis unit vector. This operation underlies resolving a force into components along two perpendicular axes (inclined plane problems), finding the component of velocity along a direction of travel, and computing work done by a force (W = F·d is itself a projection-based dot product). The remainder, a minus its projection onto b, gives the component of a perpendicular to b — useful for splitting any vector into 'along' and 'across' parts.

What's the key thing to understand about Position Vectors and Displacement Vectors?

A position vector describes the location of a point relative to a fixed origin — for point P at coordinates (3, 4), its position vector is simply (3, 4), or 3i + 4j. A displacement vector describes the change in position between two points, calculated by subtracting one position vector from another: the displacement from point A(1, 2) to point B(4, 6) is B - A = (4-1, 6-2) = (3, 4). This distinction matters because physical quantities like velocity and force are usually displacement-type vectors (they describe a change or direction, not an absolute location), while position vectors are what you use to describe where an object actually is in space. Many vector problems — like finding the midpoint of a line segment, (A + B)/2, or dividing a segment in a given ratio — rely on manipulating position vectors directly rather than displacement vectors.

What should I know about a Vector Operations Quick Reference?

The core vector formulas used throughout this page, gathered in one place for quick lookup:

What should I know about a Worked Example Combining Multiple Vector Operations?

As a fully worked 3D example, take a = (2, 3, 6) and b = (1, 0, 2). Magnitudes: |a| = √(4+9+36) = √49 = 7. |b| = √(1+0+4) = √5 ≈ 2.236. Dot product: a·b = 2×1 + 3×0 + 6×2 = 2+0+12 = 14. Angle between them: θ = cos⁻¹(14/(7×2.236)) = cos⁻¹(0.8945) ≈ 26.6°. Cross product: a×b = (3×2−6×0, 6×1−2×2, 2×0−3×1) = (6, 2, −3). Checking this is perpendicular to both: a·(a×b) = 2×6+3×2+6×(−3) = 12+6−18 = 0 ✓, and b·(a×b) = 1×6+0×2+2×(−3) = 6−6 = 0 ✓, confirming the cross product is perpendicular to both original vectors. The magnitude of the cross product, |a×b| = √(36+4+9) = √49 = 7, also matches |a||b|sinθ = 7×2.236×sin(26.6°) ≈ 7.0, confirming the area of the parallelogram spanned by a and b is 7 square units.

What should I know about the Vector Equation of a Line?

A vector equation describes every point on a straight line using a single parameter. The general form is r = a + t·b, where a is the position vector of any one known point on the line, b is a direction vector parallel to the line, and t is a scalar parameter that sweeps r through every point on the line as it ranges over all real numbers. For example, a line through the point (1, 2, 0) in the direction (2, −1, 3) is written r = (1,2,0) + t(2,−1,3), so the point at t=2 is (1+4, 2−2, 0+6) = (5, 0, 6). This vector form generalises to any number of dimensions and is the standard way lines are represented in 3D computer graphics, robotics path-planning, and mechanics problems involving the trajectory of a particle moving at constant velocity, where the direction vector b is simply the velocity and t is time. Two lines in 3D are parallel if their direction vectors are scalar multiples of each other, and unlike in 2D, two non-parallel lines in 3D generally don't intersect at all — they're 'skew' lines, passing near each other without crossing.

What should I know about Vectors in Equilibrium (Resultant Force)?

A common applied problem asks whether a set of forces leaves an object in equilibrium (no net force, so no acceleration) — answered simply by adding all the force vectors and checking whether the resultant is the zero vector. As a worked example, three forces act on a point: F1=(5,2), F2=(−3,4), F3=(−2,−6). Summing component-wise: resultant = (5−3−2, 2+4−6) = (0,0). Since the resultant is exactly the zero vector, the point is in equilibrium under these three forces. If the resultant had come out non-zero, say (1,0), that remaining vector would represent the net unbalanced force, and by Newton's second law the object would accelerate in that direction with magnitude a=F/m. This check — summing all force vectors component-wise and testing for zero — is the standard first step in any statics problem, from a hanging sign supported by two cables to a bridge truss carrying multiple loads.

What should I know about Direction Cosines?

A vector's direction can also be described using direction cosines — the cosines of the angles the vector makes with each coordinate axis. For a=(3,4,12), with magnitude |a|=√(9+16+144)=√169=13, the direction cosines are cosα=3/13≈0.231, cosβ=4/13≈0.308, cosγ=12/13≈0.923 (relative to the x, y, and z axes respectively). A key property is that the sum of their squares always equals 1: 0.231²+0.308²+0.923²≈1.0 ✓, which follows directly from the Pythagorean definition of magnitude and offers a quick way to check a set of direction cosines for consistency.