Vector Calculator
Solve vector problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations. Includes formulas and worked examples.
Formula
Operations: a+b, a-b, k*a, a.b, a x b, |a|, proj_b(a)
This calculator performs all standard vector operations including addition, subtraction, scalar multiplication, dot product, cross product, magnitude calculation, unit vector normalization, and vector projection.
Worked Examples
Example 1: Complete 3D Vector Analysis
Problem: Given a = (3, -1, 4) and b = (2, 5, -2), find all operations.
Solution: Magnitudes: |a| = sqrt(9+1+16) = sqrt(26) = 5.099, |b| = sqrt(4+25+4) = sqrt(33) = 5.745\nSum = (5, 4, 2), Diff = (1, -6, 6)\nDot = 6 + (-5) + (-8) = -7\nCross = ((-1)(-2)-(4)(5), (4)(2)-(3)(-2), (3)(5)-(-1)(2)) = (-18, 14, 17)\nAngle = arccos(-7 / (5.099 * 5.745)) = arccos(-0.2389) = 103.82 degrees
Result: Dot = -7 | Cross = (-18, 14, 17) | Angle = 103.82 deg
Example 2: Scalar Multiplication and Projection
Problem: Scale vector a = (3, -1, 4) by 3 and project a onto b = (2, 5, -2).
Solution: Scaled: 3*(3, -1, 4) = (9, -3, 12), magnitude = sqrt(81+9+144) = sqrt(234) = 15.297\nDot product a.b = -7\n|b|^2 = 33\nScalar projection = -7/5.745 = -1.2185\nVector projection = (-7/33)*(2,5,-2) = (-0.4242, -1.0606, 0.4242)\nPerpendicular component = a - proj = (3.424, 0.061, 3.576)
Result: 3a = (9, -3, 12) | Scalar proj = -1.219 | Vec proj = (-0.424, -1.061, 0.424)
Frequently Asked Questions
What is a vector and how is it different from a scalar?
A vector is a mathematical quantity that has both magnitude (size) and direction, while a scalar has only magnitude. Vectors are represented by ordered sets of numbers called components, such as (3, 4, 5) in three dimensions. Examples of vectors include displacement, velocity, force, and acceleration. Examples of scalars include temperature, mass, speed, and energy. Vectors follow special rules for addition and multiplication that account for direction, whereas scalars follow ordinary arithmetic. In notation, vectors are often written in boldface or with arrows above them to distinguish them from scalars. Understanding the vector-scalar distinction is fundamental to physics and engineering.
How do you calculate the magnitude of a vector?
The magnitude (or length or norm) of a vector v = (vx, vy, vz) is calculated using the formula |v| = sqrt(vx^2 + vy^2 + vz^2), which is a direct application of the Pythagorean theorem extended to three dimensions. In 2D, it simplifies to |v| = sqrt(vx^2 + vy^2). The magnitude is always a non-negative scalar value, and it equals zero only for the zero vector. Magnitude represents the physical size of the quantity: for a velocity vector, it gives the speed; for a force vector, it gives the force intensity. The squared magnitude (vx^2 + vy^2 + vz^2) is often used in computations instead to avoid the computationally expensive square root operation.
What is vector projection and what are its applications?
Vector projection decomposes vector a into two components: one parallel to vector b and one perpendicular to b. The scalar projection (component) is |a|cos(theta) = (a dot b)/|b|. The vector projection onto b is ((a dot b)/|b|^2) * b. The perpendicular component is a minus the projection. Applications include: resolving forces into components along and perpendicular to a surface in physics, finding the closest point on a line to a given point in geometry, decomposing velocity into radial and tangential components in orbital mechanics, shadow calculations in computer graphics, and signal decomposition in engineering. The Gram-Schmidt process uses repeated projections to orthogonalize a set of vectors.
Can I use Vector Calculator on a mobile device?
Yes. All calculators on NovaCalculator are fully responsive and work on smartphones, tablets, and desktops. The layout adapts automatically to your screen size.
Can I use the results for professional or academic purposes?
You may use the results for reference and educational purposes. For professional reports, academic papers, or critical decisions, we recommend verifying outputs against peer-reviewed sources or consulting a qualified expert in the relevant field.
What formula does Vector Calculator use?
The formula used is described in the Formula section on this page. It is based on widely accepted standards in the relevant field. If you need a specific reference or citation, the References section provides links to authoritative sources.