Choose one level below and stay focused on that material only. Each level hides the others so the page feels like one workspace instead of a long stack of mixed topics.
Only one level is shown at a time so you can stay focused on the right concepts and labs.
Scalars, components, arithmetic, and direction in the coordinate plane.
A scalar has magnitude only (temperature, mass, speed). A vector has both magnitude and direction (velocity, force, displacement).
Vectors can be written as arrows, ordered pairs, or unit-vector form. Common notations: \(\vec{v}\), \(\mathbf{v}\), or component form \(\langle v_x, v_y \rangle\).
Drag the endpoint to change the vector, then read the updated components and angle below.
A 2-D vector is defined by its \(x\) and \(y\) components. Drag the endpoint below to explore how the components, magnitude, and direction angle update in real time.
Drag the vectors on the canvas or edit the input boxes below to test addition, subtraction, and scalar multiplication instantly.
Relative motion, dot product, projection, and basis-vector thinking.
Change the two speeds to see how independent motion vectors combine into one path.
Relative velocity combines two velocity vectors. A classic example: a boat crossing a river. The boat's velocity (north) and the river current (east) add to produce the resultant.
Drag both vectors until the dot product approaches zero, then watch the projection shrink.
\(\vec{u} \cdot \vec{v} = u_x v_x + u_y v_y = \|\vec{u}\|\|\vec{v}\|\cos\theta\). When \(\vec{u} \cdot \vec{v} = 0\), the vectors are orthogonal (perpendicular).
Scalar projection:
\(\text{comp}_{\vec{v}}\vec{u} = \dfrac{\vec{u}\cdot\vec{v}}{\|\vec{v}\|}\)
Vector projection:
\(\text{proj}_{\vec{v}}\vec{u} = \dfrac{\vec{u}\cdot\vec{v}}{\|\vec{v}\|^2}\,\vec{v}\)
Tip: The Dot Product interactive above also visualises the projection vector in purple.
The standard basis in 2-D is \(\hat{i} = \langle 1,0 \rangle\) and \(\hat{j} = \langle 0,1 \rangle\). Any vector can be expressed as a linear combination: \(\vec{v} = v_x\hat{i} + v_y\hat{j}\).
3-D vectors, cross products, line and plane equations, and motion functions.
Everything extends naturally: \(\vec{v} = \langle x, y, z \rangle\), \(\|\vec{v}\| = \sqrt{x^2+y^2+z^2}\). The standard basis adds \(\hat{k} = \langle 0,0,1 \rangle\).
Adjust the 3-D inputs and compare the resulting perpendicular vector in the readout.
\(\vec{u} \times \vec{v} = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\u_x&u_y&u_z\\v_x&v_y&v_z\end{vmatrix}\). The result is perpendicular to both inputs and its magnitude equals the area of the parallelogram they span.
Line: \(\vec{r}(t) = \vec{r}_0 + t\,\vec{d}\) where \(\vec{d}\) is the direction vector.
Plane: \(\vec{n} \cdot (\vec{r} - \vec{r}_0) = 0\) where \(\vec{n}\) is the normal.
Set launch speed and angle, then animate the path to connect vectors with motion.
A vector-valued function \(\vec{r}(t) = \langle x(t),\;y(t) \rangle\) traces a curve. Its derivative \(\vec{r}'(t)\) gives the velocity vector; the second derivative gives acceleration. Projectile motion is a natural application:
\[\vec{r}(t) = \langle v_0\cos\theta\;t,\;\; v_0\sin\theta\;t - \tfrac{1}{2}gt^2 \rangle\]