Vector Calculator
Solve vector problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations. Includes formulas and worked examples.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Vector Calculator
Calculator
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Formula: Operations: a+b, a-b, k*a, a.b, a x b, |a|, proj_b(a)
Worked example — Dot = -7 | Cross = (-18, 14, 17) | Angle = 103.82 deg
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 Sum = (5, 4, 2), Diff = (1, -6, 6) Dot = 6 + (-5) + (-8) = -7 Cross = ((-1)(-2)-(4)(5), (4)(2)-(3)(-2), (3)(5)-(-1)(2)) = (-18, 14, 17) Angle = 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 Dot product a.b = -7 |b|^2 = 33 Scalar projection = -7/5.745 = -1.2185 Vector projection = (-7/33)*(2,5,-2) = (-0.4242, -1.0606, 0.4242) Perpendicular 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.
What are all the different operations you can perform on vectors?
The main vector operations include addition (combining two vectors component-wise), subtraction (finding the difference vector), scalar multiplication (scaling a vector by a number), dot product (multiplying two vectors to get a scalar result), cross product (multiplying two vectors to get a perpendicular vector), and normalization (converting to a unit vector). Additional operations include vector projection (finding the component of one vector along another), finding the angle between vectors using the dot product formula, calculating the triple scalar product for volume calculations, and computing linear combinations. Each operation has specific geometric and physical interpretations that make vectors one of the most powerful tools in applied mathematics.
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 scalar multiplication and what effect does it have?
Scalar multiplication multiplies each component of a vector by a single number (scalar): k*(vx, vy, vz) = (k*vx, k*vy, k*vz). This operation scales the vector, changing its magnitude while preserving or reversing its direction. If k is positive, the vector keeps its direction but its length is multiplied by k. If k is negative, the vector reverses direction and its length is multiplied by |k|. If k equals zero, the result is the zero vector. Scalar multiplication is distributive over vector addition: k*(a + b) = k*a + k*b. In physics, multiplying a velocity vector by time gives displacement, and multiplying a unit direction vector by a force magnitude gives a force vector.
When would you use the dot product versus the cross product?
Use the dot product when you need to find the angle between vectors, compute projections, check orthogonality, or calculate work (force dot displacement). The dot product produces a scalar and works in any number of dimensions. Use the cross product when you need to find a vector perpendicular to two given vectors, compute area of a parallelogram, calculate torque (position cross force), find surface normals, or determine the orientation of a turning motion. The cross product produces a vector and is only defined in three dimensions (and seven dimensions mathematically). In summary, dot product tells you about alignment and the cross product tells you about perpendicularity and rotation.
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.
How are vectors used in computer graphics and 3D rendering?
Vectors are the mathematical backbone of computer graphics. Position vectors define locations of vertices and objects. Normal vectors perpendicular to surfaces determine how light reflects, enabling realistic shading through the dot product between normal and light direction vectors. Direction vectors define camera orientation and ray directions in ray tracing. Velocity vectors animate object movement. Tangent and bitangent vectors along with normals form the TBN matrix for normal mapping. Transformation matrices multiply position vectors to achieve rotation, scaling, and translation. Cross products create perpendicular vectors for constructing coordinate frames. Bezier curves use vector-valued parametric equations for smooth path generation.
What is the parallelogram law and how does it relate to vectors?
The parallelogram law states that the sum of the squares of the diagonals of a parallelogram equals the sum of the squares of all four sides. In vector terms: |a + b|^2 + |a - b|^2 = 2(|a|^2 + |b|^2). Geometrically, when two vectors a and b originate from the same point, they form two sides of a parallelogram. The diagonal from the origin to the opposite corner is a + b, and the other diagonal is a - b. This law provides a useful identity for checking calculations and has deep connections to inner product spaces. It also gives an alternative method for vector addition called the parallelogram method, which is equivalent to the tip-to-tail method.
What are linearly independent vectors and why do they matter?
Vectors are linearly independent if none of them can be expressed as a linear combination (sum of scalar multiples) of the others. In 3D space, three linearly independent vectors form a basis, meaning any vector in the space can be uniquely expressed as a combination of these three vectors. The standard basis vectors i, j, k are the most common example. Two vectors are linearly dependent if and only if they are parallel (one is a scalar multiple of the other). Three 3D vectors are linearly dependent if they all lie in the same plane. Linear independence is fundamental in linear algebra for solving systems of equations, spanning vector spaces, and determining matrix rank.
How do vectors extend to higher dimensions beyond 3D?
Vectors generalize seamlessly to any number of dimensions. An n-dimensional vector has n components: v = (v1, v2, ..., vn). All fundamental operations extend naturally: addition adds corresponding components, scalar multiplication scales all components, the dot product sums products of corresponding components, and magnitude uses the square root of the sum of squared components. However, the cross product as defined does not generalize beyond 3D (except to 7D). In data science, vectors with hundreds or thousands of dimensions represent data points in feature spaces. Word embeddings in natural language processing use 300-dimensional vectors. Machine learning models operate on high-dimensional vectors, and the geometric intuition from 2D and 3D carries over conceptually.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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