CLASSICAL-MECHANICS · Interactive Practice

Same Vector, New Numbers: Components Under Rotated Axes

IKey Formulas

Formula	Name	Description
$\hat{i}' = \hat{i}\cos\theta + \hat{j}\sin\theta$	Rotated unit vector	New x-axis basis in old basis
$\hat{j}' = \hat{j}\cos\theta - \hat{i}\sin\theta$	Rotated unit vector	New y-axis basis in old basis
$x' = x\cos\theta + y\sin\theta$	Component transform	New x-component
$y' = -x\sin\theta + y\cos\theta$	Component transform	New y-component (note the minus sign)
$\lvert \vec{R}\rvert = \sqrt{x^2+y^2} = \sqrt{x'^2+y'^2}$	Magnitude invariance	Length is the same in every frame

IIVisualization 1 — Components as Projections

Rotating the axes rewrites a fixed arrow's components — because $x'$ and $y'$ are its projections onto the tilted axes.

Axis rotation θ 30°

💡 Challenge: find the rotation $\theta$ that drives $x'$ to $0$ — the arrow then lies entirely along the new $y'$ -axis.

IIIVisualization 2 — Magnitude Is Invariant

Each component swings wildly as the axes turn, so what survives every rotation?

Vector x 5 Vector y −3 θ 60°

💡 Why it matters: length is a property of the vector itself, not of the frame — the same holds for the dot product of any two vectors.

IVQuiz Questions

Question 1

A vector has components $x = 3$ , $y = 4$ in system $S$ . The axes are rotated by $\theta = 90°$ to form system $S'$ .

Using $x' = x\cos\theta + y\sin\theta$ , what is $x'$ ?

✅ Correct! With θ = 90°, x' = 3(0) + 4(1) = 4.

❌ Not quite. Remember cos 90° = 0 and sin 90° = 1, so the cosine term vanishes.

Show solution

Solution:

With $\theta = 90°$ : $\cos 90° = 0$ and $\sin 90° = 1$ .

$x' = x\cos\theta + y\sin\theta = 3(0) + 4(1) = 4$

After a $90°$ rotation, the new x-axis points along the old y-axis, so the new x-component picks up the old y-value.

Question 2

The most common error in this derivation involves a sign.

True or False: The correct transformation for the new y-component is $y' = x\sin\theta + y\cos\theta$ .

✅ Correct! The correct form is y' = −x sin θ + y cos θ — the minus sign is essential.

❌ Not quite. Watch the sign: the x sin θ term must be negative.

Show solution

Solution:

The correct relation has a minus sign on the $x\sin\theta$ term:

$y' = -x\sin\theta + y\cos\theta$

This minus sign appears because the new $y'$ -axis tilts away from the old $x$ -direction. The statement as written omits the minus sign, so it is False.

Question 3

A vector $\vec{A}$ has components $A_x = 5$ , $A_y = -3$ in system $S$ . The axes are rotated by $\theta = 60°$ to form system $S'$ , giving new components $A_x' \approx -0.10$ and $A_y' \approx -5.83$ .

What is the magnitude $\lvert\vec{A}\rvert$ measured in system $S'$ ?

✅ Correct! Magnitude is invariant: √(25 + 9) = √34 in both frames.

❌ Not quite. Length doesn't change under rotation — compute √(5² + (−3)²) in the original frame.

Show solution

Solution:

Magnitude is invariant under rotation, so compute it in either frame.

In $S$ : $\lvert\vec{A}\rvert = \sqrt{A_x^2 + A_y^2} = \sqrt{25 + 9} = \sqrt{34} \approx 5.83$

In $S'$ : $\lvert\vec{A}\rvert = \sqrt{(-0.10)^2 + (-5.83)^2} \approx \sqrt{34} \approx 5.83$

The components look completely different, but the length is identical in both frames.

Question 4

Consider the conceptual meaning of rotating the coordinate axes.

True or False: Rotating the axes by an angle $+\theta$ relabels the vector's components in exactly the same way as keeping the axes fixed and rotating the vector itself by $-\theta$ .

✅ Correct! A passive rotation of the axes by +θ matches an active rotation of the vector by −θ.

❌ Not quite. Rotating the axes one way relabels components just like rotating the vector the opposite way.

Show solution

Solution:

This is the distinction between a passive rotation (rotating the axes) and an active rotation (rotating the vector).

Rotating the axes by $+\theta$ produces the same new component numbers as leaving the axes fixed and rotating the vector by $-\theta$ . In this video nothing physical moved — only our description of the fixed arrow changed. So the statement is True.

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