Undergraduate · Orthogonal projections · 280 of 650
Orthogonal projections · Vector horizontal a=1.5; Vector vertical b=2.5
Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length. Givens: Vector horizontal a=1.5; Vector vertical b=2.5.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Let v=(1.5,2.5) and project it orthogonally onto the line with unit direction u=(cos θ,sin θ). Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length.
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A starting point
Compute c=v·u first, then the projected vector cu. Subtract it from v to obtain the perpendicular residual.
Work through the reasoning
Step 1
Identify the model and target
Let v=(1.5,2.5) and project it orthogonally onto the line with unit direction u=(cos θ,sin θ). Use θ=6π/5 radians. The governing relation is projᵤv=(v·u)u; |u|=1. Compute c=v·u first, then the projected vector cu. Subtract it from v to obtain the perpendicular residual.
Step 2
Substitute and calculate
The unit direction is (-0.809017,-0.587785). Compute v·u=(1.5)(-0.809017)+(2.5)(-0.587785)=-2.682989, then multiply that scalar back into the direction to get (2.170583,1.577021). Subtract this from (1.5,2.5); the remaining vector has length 1.140865.
Step 3
Check the mathematical meaning
The residual is (-0.670583,0.922979). Its dot product with u is zero; also |v|²=8.5 equals |projection|²+|residual|². Projection x: 2.171; Projection y: 1.577; Signed component: -2.683; Residual length: 1.141. Decimal values are rounded, so use unrounded intermediate values.
The answer
The unit direction is (-0.809017,-0.587785). Compute v·u=(1.5)(-0.809017)+(2.5)(-0.587785)=-2.682989, then multiply that scalar back into the direction to get (2.170583,1.577021). Subtract this from (1.5,2.5); the remaining vector has length 1.140865. The residual is (-0.670583,0.922979). Its dot product with u is zero; also |v|²=8.5 equals |projection|²+|residual|². Animation check: Projection x: 2.171; Projection y: 1.577; Signed component: -2.683; Residual length: 1.141. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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The mathematical idea
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. Let v=(1.5,2.5) and project it orthogonally onto the line with unit direction u=(cos θ,sin θ). Use θ=6π/5 radians. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
projᵤv=(v·u)u; |u|=1
03 · Reflect and transfer
Explain what changes and why.
Reverse the unit direction u. Why does the signed component reverse while the projected vector stays the same?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.