Vectors
Vectors is a free interactive simulation from StudyFi Lab. It runs right in your browser, with no sign-up and nothing to install.
Add, scale and multiply vectors in 2D and 3D
Experiments
Basics
A vector has a size and a direction. We write it with coordinates a = (a₁, a₂, a₃) or with the unit vectors of the axes as a = a₁i + a₂j + a₃k. Its length comes from Pythagoras, |a| = √(a₁² + a₂² + a₃²), and the unit vector â = a / |a| keeps the direction with length 1. The vector from point A to point B is the difference B − A.
Try this: Drag the tip to (3, 4) and check that |a| = 5. Then switch on “From two points” and move both points by the same amount: the vector stays the same.
Adding vectors
Vectors add component by component: a + b = (a₁ + b₁, a₂ + b₂). On paper you move b to the tip of a (triangle rule), or you build a parallelogram from both vectors and the sum is its diagonal. The difference a − b is the sum a + (−b): the vector that leads from the tip of b to the tip of a.
Try this: Press “Place b at the tip of a” and watch the triangle form. Then switch to the parallelogram and set b = −a: the sum becomes the zero vector.
Scalar multiple
The multiple k·a multiplies every component by k, so it lies on the same line as a: the vectors are collinear. The length becomes |k|·|a|. For k > 0 the direction stays, for k < 0 it flips to the opposite one, and k = 0 gives the zero vector.
Try this: Move k from 2 to −2 and watch the arrow flip through the origin while its length stays the same.
Dot product
The dot product is a number: a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a|·|b|·cos φ. It is positive when the vectors form an acute angle, zero at a right angle and negative at an obtuse one, so a·b = 0 means the vectors are perpendicular. The projection of b onto the direction of a has length a·b / |a|.
Try this: Set a = (4, 1) and look for a b with a·b = 0. Can you find (−1, 4)? Always guess the sign first.
Cross product
The cross product a × b is a vector perpendicular to both a and b. The right-hand rule gives its direction, and its length |a × b| = |a|·|b|·sin φ equals the area of the parallelogram spanned by a and b. It is zero for parallel vectors, and swapping the factors flips it: b × a = −(a × b).
Try this: Set a = (3, 0, 0) and b = (0, 2, 0): a × b points along the z axis with length 6, just like the 3 × 2 rectangle. Then tilt b towards a and watch the area shrink.
Scalar triple product
The scalar triple product (a × b)·c is a number equal to the determinant of the components of a, b and c. Its absolute value is the volume of the parallelepiped the three vectors span. When it is zero, the parallelepiped is flat: all three vectors lie in one plane (they are coplanar).
Try this: Set c = a − b, which is c = (3, −1, −1) for the starting a and b: the volume drops to zero. Then raise c₃ and watch the solid puff up.
Frequently asked questions
What is Vectors?
Vectors is a free interactive simulation from StudyFi Lab. It runs right in your browser, with no sign-up and nothing to install. Add, scale and multiply vectors in 2D and 3D
Is Vectors free?
Yes. Vectors is completely free. It works in the browser on computers, tablets and phones, and you do not need an account.
What can I explore in Vectors?
Basics, Adding vectors, Scalar multiple, Dot product, Cross product, Scalar triple product.
What does the “Basics” experiment show?
A vector has a size and a direction. We write it with coordinates a = (a₁, a₂, a₃) or with the unit vectors of the axes as a = a₁i + a₂j + a₃k. Its length comes from Pythagoras, |a| = √(a₁² + a₂² + a₃²), and the unit vector â = a / |a| keeps the direction with length 1. The vector from point A to point B is the difference B − A.
What does the “Adding vectors” experiment show?
Vectors add component by component: a + b = (a₁ + b₁, a₂ + b₂). On paper you move b to the tip of a (triangle rule), or you build a parallelogram from both vectors and the sum is its diagonal. The difference a − b is the sum a + (−b): the vector that leads from the tip of b to the tip of a.
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