Mon Prof Augmenté
Two Vectors on a Grid
Age 16 · Maths
Two arrows start at the same point on a grid, with an open angle between them. The dot product combines their lengths and the cosine of that angle, and it also gives the length of one arrow’s projection onto the other.
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Age 16: open this week
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In the picture
- Shared starting point: the origin where both arrows begin.
- Longer arrow: its length is the norm of one vector.
- Shorter arrow: its length is the norm of the other vector.
- Curved arc: the angle between the two arrows.
- Shaded stretch along the longer arrow: the projection length of the shorter arrow onto the longer one.
Examples
- Pulling a sled with a rope at an angle: only the part of the pull along the sled’s path moves it forward.
- A solar panel tilted away from direct sunlight receives less energy because the angle changes the projection of the sunlight.
Key takeaways
- The dot product is zero exactly when two nonzero vectors are perpendicular.
- For two vectors, cos θ = (a · b) / (|a| |b|).
Quiz: Two Vectors on a Grid (Age 16)
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For two nonzero vectors, when is their dot product equal to zero?
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Which formula gives the cosine of the angle θ between two vectors a and b?
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Besides combining the vectors' lengths and the cosine of their angle, what else does the dot product give?
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