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Grade 12 · Vector components and angles

Vector components and angles: Equal magnitudes with negative horizontal

Vector components and angles: investigate equal magnitudes with negative horizontal with vector length r = 2; final angle (degrees) = 135.

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

Watch the relationship

Paused
Vector components and angles: Equal magnitudes with negative horizontal. Angle in degrees: 0. Dot product with (1,0): 2. Vertical component: 0. Vector length: 2A signed component along a unit axisxy0r = 2θ = 0°Equal axis scales · coordinates in the readouts
The comparison direction has unit length. Coordinates are Euclidean and the angle lies between zero and 180 degrees. A dot product is a scalar, whereas the projection vector would be that scalar times (1,0).

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.

Angle in degrees
0
Dot product with (1,0)
2
Vertical component
0
Vector length
2

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From experiment to screen.

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Understand what you are seeing

The idea behind the motion.

The dot product with a unit direction measures signed component along that direction. A positive component points with the axis, a zero component is perpendicular, and a negative component points against it. The vector length remains fixed while its horizontal contribution changes. This investigation starts with Vector length r = 2; Final angle (degrees) = 135. 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

v=r(cos θ,sin θ); v·(1,0)=r cos θ

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

    Identify the fixed unit direction (1,0) and the rotating vector of length r. The starting case is “Equal magnitudes with negative horizontal.”

  2. STEP 2

    Follow the changing quantity

    Rotate through the selected angle and read the horizontal projection rather than the full vector length.

  3. STEP 3

    Explain and test the result

    Compare acute, right, and obtuse angles. Use the sign of cosine to explain the sign of the dot product and connect the motion to polar-circle coordinates.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Vector length r = 2; Final angle (degrees) = 135. Pause the timeline at 60%. Given angle in degrees = 81, calculate dot product with (1,0), vertical component, vector 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

At θ=81°, the coordinates are (2 cos(81°), 2 sin(81°))=(0.312869,1.975377). Their squared sum is 4, and the dot product with (1,0) selects the first coordinate. Results: Dot product with (1,0): 0.313; Vertical component: 1.975; Vector length: 2. 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.

Grade 12 · Geometry

Recognize a circle hidden in a polar equation

Translate a polar curve into a Cartesian equation.