Use the simulator above to explore how vectors add, both graphically and by components. Build up to three vectors in polar form (magnitude and angle) or component form, place them tip to tail or from the origin, and watch the resultant update live as you drag, resize, or retype them. Display options let you show each vector's x and y components and choose whether angles are measured from the x-axis or the y-axis, so students can see why sine and cosine swap roles. When students are ready, Match Resultant mode generates unlimited targets to build toward, so they can practice the skill until they reach mastery.

How to Use the Vector Addition Simulator
 

Choosing a Mode

• Pick Explore to build vectors freely and see how they add. Pick Match Resultant for a challenge where you build vectors to hit a target.
  Note for teachers: Start students in Explore. Once they can predict where the resultant will land before they look, move them to Match Resultant.
   

Placement

• Tip to Tail chains each vector's tail to the previous vector's tip, and the resultant R runs from the start of the first vector to the tip of the last.

• From Origin puts every vector's tail at the origin, which is how you compare their sizes and directions side by side.
  Mastery tip: Switch between the two placements with the same vectors. The resultant is identical, and only the picture changes.
   

Vectors and Vector Input

• Press Add Vector to add up to three vectors, − to remove one, and Clear to start over. Each vector is a row labeled A, B, or C, and the resultant row R updates live.

• Choose Polar (|v|, θ) to enter a magnitude and an angle, or Components (x, y) to enter the x and y parts directly.

• You can also drag a vector's head to resize it and drag its body to move it. Use two fingers to pan and pinch to zoom on a touch screen.
   

Display

• Show grid and Show labels & angles control what is drawn on the canvas.

• Show per-vector components and Show resultant components draw the x and y parts of each vector and of R.

• Choose Light or Dark for the theme, and use the Zoom slider (0.4× to 5×) or the 1:1 button to reset pan and zoom. The fullscreen button gives the canvas the whole screen.
   

Components

• Order sets how the components are laid out. In x then y, the x component starts at the tail and the y component starts at the end of it. y then x reverses that. Show both draws both paths so you can see they reach the same tip.

• Angle sets which angle is marked. θ from +x measures from the x-axis, φ from +y measures from the y-axis, and Both shows both. The two angles are related by φ = 90° − θ.
  Mastery tip: Use Show both with Angle set to Both, then compare the equations side by side. Ax = A cos θ is the same number as Ax = A sin φ, because cos θ and sin φ are equal when the angles are complementary.
  Note for teachers: This is a quick way to show that adding components in either order gives the same vector, and that sine and cosine swap roles when you measure from the other axis.
   

Component equations (Readout Panel)

• Use the tabs A, B, C, and R to see the equations for each vector and for the resultant. Each tab shows the section From +x axis (θ), the section From +y axis (φ), and a Sum section with equations like Rx = Ax + Bx.

• You can collapse the panel when you want more room for the canvas.
   

Match Resultant (Challenge Mode)

• Press Generate Challenge to get a target. Set the Number of vectors to 1, 2, or 3. Constrain to axes only restricts your vectors to point along the axes, so it does not apply when the target is diagonal and you only have one vector.

• Build your vectors under Your Vectors, then press Submit. The tool tells you whether your resultant matches the target. Click anywhere to try again or start a new challenge.
  Mastery tip: Work out the target's x and y components first, then choose vectors whose components add up to them.