Grade 12 · Vector components and angles · 95 of 600
Vector components and angles · Vector length r=3; Final angle (degrees)=90
Evaluate at the signed angle θ=54° (three fifths of that final angle). Find its dot product with (1,0), its vertical component, and its length. Givens: Vector length r=3; Final angle (degrees)=90.
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
A vector has length r=3; its angle is measured counterclockwise from the positive horizontal axis. The animation's final angle is 90°. Evaluate at the signed angle θ=54° (three fifths of that final angle). Find its dot product with (1,0), its vertical component, and its length.
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A starting point
Use (r cos θ,r sin θ) with degrees. The dot product with (1,0) selects the first component.
Work through the reasoning
Step 1
Identify the model and target
A vector has length r=3; its angle is measured counterclockwise from the positive horizontal axis. The animation's final angle is 90°. Evaluate at the signed angle θ=54° (three fifths of that final angle). The governing relation is v=r(cos θ,sin θ); v·(1,0)=r cos θ. Use (r cos θ,r sin θ) with degrees. The dot product with (1,0) selects the first component.
Step 2
Substitute and calculate
At θ=54°, the coordinates are (3 cos(54°), 3 sin(54°))=(1.763356,2.427051). Their squared sum is 9, and the dot product with (1,0) selects the first coordinate.
Step 3
Check the mathematical meaning
The squared components sum to r²=9. Their signs must match the quadrant of θ=54°. A negative angle rotates clockwise. Angle in degrees: 54; Dot product with (1,0): 1.763; Vertical component: 2.427; Vector length: 3. Decimal values are rounded, so use unrounded intermediate values.
The answer
At θ=54°, the coordinates are (3 cos(54°), 3 sin(54°))=(1.763356,2.427051). Their squared sum is 9, and the dot product with (1,0) selects the first coordinate. The squared components sum to r²=9. Their signs must match the quadrant of θ=54°. A negative angle rotates clockwise. Animation check: Angle in degrees: 54; Dot product with (1,0): 1.763; Vertical component: 2.427; Vector length: 3. 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.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
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The mathematical idea
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. A vector has length r=3; its angle is measured counterclockwise from the positive horizontal axis. The animation's final angle is 90°. Evaluate at the signed angle θ=54° (three fifths of that final angle). The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
v=r(cos θ,sin θ); v·(1,0)=r cos θ
03 · Reflect and transfer
Explain what changes and why.
Increase the angle by 180°. What changes sign, and which quantity stays unchanged?
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.