How To Do Vectors In Desmos

How to Do Vectors in Desmos: A Step‑by‑Step Guide for Students and Educators

Desmos is a powerful, free online graphing calculator that lets you visualize mathematical concepts instantly. When it comes to vectors, Desmos makes it easy to plot, add, subtract, scale, and even animate them—turning abstract ideas into concrete pictures you can manipulate in real time. This guide walks you through everything you need to know to work with vectors in Desmos, from basic entry to advanced applications, so you can use the tool confidently in homework, lectures, or personal exploration.

Getting Started with Desmos

Before diving into vectors, make sure you have a Desmos account (optional but useful for saving work) and a blank graph open. The interface consists of an expression list on the left and a coordinate plane on the right. You can type any mathematical expression directly into the list, and Desmos will render it instantly.

Key terms to know

• Vector: a quantity with both magnitude and direction, often written as ⟨x, y⟩ in 2‑D or ⟨x, y, z⟩ in 3‑D.
• Component form: the representation of a vector by its horizontal and vertical (or depth) parts. - Scalar multiplication: multiplying a vector by a real number changes its length but not its direction (unless the scalar is negative, which flips direction).

Entering a Vector in Desmos

Desmos does not have a dedicated “vector” button, but you can represent vectors using points, arrows, or parametric lines. The simplest way is to treat a vector as a point whose coordinates are its components, then draw an arrow from the origin (or any starting point) to that point.

1. Basic 2‑D Vector from the Origin1. In the expression list, type:

A = (0,0)
B = ⟨3, 4⟩

2. Desmos will automatically interpret ⟨3, 4⟩ as a point (3, 4).
3. To draw the vector, add a line or arrow:

   Vector = segment(A, B)
   

   or for an arrowhead:

   Vector = polygon(A, B, (B.x + 0.2*(B.y-A.y), B.y - 0.2*(B.x-A.x)), (B.x - 0.2*(B.y-A.y), B.y + 0.2*(B.x-A.x)))
   

   (Desmos also offers a built‑in arrow command: arrow(A, B) in newer versions.)

The segment (or arrow) now appears on the graph, representing the vector ⟨3, 4⟩ starting at the origin.

2. Vector with a Custom Starting Point

If you need a vector that does not start at (0,0), define both the tail and the tip:

Tail = (1, 2)
Tip  = Tail + ⟨3, 4⟩   // results in (4, 6)
Vector = segment(Tail, Tip)

Desmos updates the vector automatically whenever you change Tail or the components.

3. 3‑D Vectors (Desmos 3‑D Calculator)

Desmos offers a separate 3‑D graphing mode (accessible via the “3‑D Graph” button). The process mirrors the 2‑D case:

A = (0,0,0)
B = ⟨2, -1, 5⟩
Vector3D = segment(A, B)

You can rotate the view with your mouse to inspect the vector from any angle.

Performing Vector Operations

Once vectors are on the graph, you can manipulate them algebraically and see the results instantly.

Addition and Subtraction

To add two vectors u and v, simply add their components:

u = ⟨2, 3⟩
v = ⟨-1, 4⟩w = u + v   // Desmos computes ⟨1, 7⟩

Plot w the same way as before (segment((0,0), w)) to see the resultant vector.

Subtraction works similarly:

diff = u - v   // ⟨3, -1⟩

Scalar Multiplication

Multiply a vector by a scalar k:

k = 2.5
scaled = k * u   // ⟨5, 7.5⟩```

If `k` is negative, the vector flips direction; Desmos will show this immediately.

### Dot Product

The dot product yields a scalar and is useful for checking orthogonality:

dot = u.x * v.x + u.y * v.y // Desmos evaluates to 10

You can display the result as a label:

label = ("u·v = " + dot)

### Cross Product (3‑D Only)

In the 3‑D calculator, compute the cross product:

cross = ⟨u.yv.z - u.zv.y, u.zv.x - u.xv.z, u.xv.y - u.yv.x⟩

Plot `cross` as a vector perpendicular to both `u` and `v`.

### Magnitude (Length)

The magnitude of a vector **v** = ⟨x, y⟩ is √(x² + y²). In Desmos:

mag = sqrt(v.x^2 + v.y^2)

You can show the length as a dynamic label that updates when you drag the vector’s tip.

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## Visualizing Vector Concepts

Desmos shines when you turn static calculations into interactive explorations.

### 1. Draggable Points

Instead of fixing components with numbers, use sliders:

a = slider(-5, 5, 0.1) // horizontal component b = slider(-5, 5, 0.1) // vertical component v = ⟨a, b⟩

Now moving the sliders changes the vector in real time, letting students see how direction and length vary.

### 2. Vector Addition Parallelogram

To illustrate the parallelogram law:

u = ⟨2, 1⟩ v = ⟨1, 3⟩ tipU = utipV = v tipSum = u + v

// Draw the parallelogram par = polygon((0,0), tipU, tipSum, tipV)

The shape updates as you adjust `u` or `v` via sliders.

### 3. Projection of One Vector onto Another

The projection of **v** onto **u** is:

proj = ( (u·v) / (u·u) ) * u

Plot both `v` and `proj` to see the shadow of `v` on the line spanned by `u`.

### 4. Animation with Time Parameter

Introduce a time variable `t` to animate a vector rotating or scaling:

t = slider(0, 2π, 0.01) r = 3 v_t =