Vectors and the Laws of Physics — Question 3

Question 3

A vector equation survives a common rotation

An experiment reports $\vec A=(2,3)$ m, $\vec B=(-1,4)$ m and $\vec C=(1,7)$ m. Its proposed relation is $\vec C=\vec A+\vec B$. Use $45^\circ$ counterclockwise rotated axes, for which $(X_{x\prime},X_{y\prime})=((X_x+X_y)/\sqrt2,(-X_x+X_y)/\sqrt2)$.

1. Check the relation in the original axes and transform all three vectors.

2. Check both primed component equations.

3. Explain why checking only one component, or transforming only C, is insufficient.

Question 3 - Solution

A vector equation survives a common rotation

Here $Q_\phi$ transforms components when axes turn counterclockwise by $\phi$; $R_\phi=Q_{-\phi}$ rotates the arrow in fixed axes. An unsubscripted $Q$ denotes the same component transformation, and $I$ is the identity.

(a) Transform every vector by the same rule.

$$\vec A+\vec B=(1,7)=\vec C;\quad\boxed{\vec A'=\tfrac{(5,1)}{\sqrt2},\ \vec B'=\tfrac{(3,5)}{\sqrt2},\ \vec C'=\tfrac{(8,6)}{\sqrt2}}\quad\mathrm m$$ Originally $(2,3)+(-1,4)=(1,7)$. The new components are $\vec A:(5/\sqrt2,1/\sqrt2)$ m, $\vec B:(3/\sqrt2,5/\sqrt2)$ m, and $\vec C:(8/\sqrt2,6/\sqrt2)$ m.

(b) Check equality along both independent axes.

$$A_x'+B_x'=\frac{5+3}{\sqrt2}=C_x',\quad A_y'+B_y'=\frac{1+5}{\sqrt2}=C_y'$$ The horizontal sum is $(5+3)/\sqrt2=8/\sqrt2$ m and the vertical sum is $(1+5)/\sqrt2=6/\sqrt2$ m. Both agree with C. Projection is linear, so a true vector sum remains true after a common change of axes.

(c) Require a complete, consistent representation.

$$Q_\phi(\vec A+\vec B)=Q_\phi\vec A+Q_\phi\vec B,\qquad C_x'=A_x'+B_x'\text{ alone leaves }C_y'\text{ unchecked}$$ One matching component permits an undetected mismatch along the perpendicular direction. Transforming C but leaving A and B in the old frame combines incompatible coordinate descriptions. Frame-independent validity means the same vector relation holds when all entries are expressed consistently.