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Undergraduate · Orthogonal projections

Orthogonal projections: A diagonal input

Orthogonal projections: investigate a diagonal input with vector horizontal a = 2; vector vertical b = 2.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

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Orthogonal projections: A diagonal input. Projection x: 2. Projection y: 0. Signed component: 2. Residual length: 2Keep a component; remove a residualxy0InputProjectionResidualEqual axis scales · coordinates in the readouts
The line passes through the origin and uses a normalized direction. Projection is Euclidean. The origin vector is allowed and projects to zero onto every line. The two coordinate scales are equal.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Projection x
2
Projection y
0
Signed component
2
Residual length
2

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Understand what you are seeing

The idea behind the motion.

Projection decomposes a vector into a part on a chosen line and a perpendicular residual. Rotating that line changes the best approximation but leaves the original vector fixed. Orthogonality of the residual, not a visual resemblance alone, certifies the least-distance projection. This investigation starts with Vector horizontal a = 2; Vector vertical b = 2. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

projᵤv=(v·u)u; |u|=1

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Read the fixed vector v=(a,b) and begin with the horizontal unit direction. The starting case is “A diagonal input.”

  2. STEP 2

    Follow the changing quantity

    Rotate the direction through one turn. Follow the projected vector and the residual that completes the original vector.

  3. STEP 3

    Explain and test the result

    Check that the residual's dot product with the unit direction is zero. Compare cases where the projection captures all or none of the vector.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Vector horizontal a = 2; Vector vertical b = 2. Pause the timeline at 60%. Given projection x = 2.26, calculate projection y, signed component, residual length. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

The unit direction is (-0.809017,-0.587785). Compute v·u=(2)(-0.809017)+(2)(-0.587785)=-2.793604, then multiply that scalar back into the direction to get (2.260074,1.64204). Subtract this from (2,2); the remaining vector has length 0.442463. Results: Projection y: 1.642; Signed component: -2.794; Residual length: 0.442. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Undergraduate · Algebra

Separate a signal from its perpendicular residual

Compute a least-distance projection onto a line.