Vectors and the Laws of Physics — Question 8

Question 8

Choose an axis along the measured vector

A vector has components $(-5,12)$ m. New perpendicular axes share the old origin and rotate counterclockwise by $\phi$. The aim is to make the new vertical component zero.

1. Find a rotation angle that also makes the new horizontal component positive.

2. Find the other axis orientation within one full turn that also makes the vertical component zero. What is its horizontal component?

3. Explain why zeroing one component does not remove the vector or change its magnitude.

Question 8 - Solution

Choose an axis along the measured vector

(a) Align positive x prime with the arrow.

$$\phi=180^\circ-\tan^{-1}(12/5)=112.620^\circ,\quad\boxed{\vec A'=(13,0)\,\mathrm m}$$ The vector has magnitude 13 m and standard angle $\phi=180^\circ-\tan^{-1}(12/5)\approx112.620^\circ$. At this orientation, $\cos\phi=-5/13$ and $\sin\phi=12/13$. Substitution gives $(A_{x\prime},A_{y\prime})=(13,0)$ m.

(b) Allow the axis to point opposite to the arrow.

$$\phi_2=\phi+180^\circ=292.620^\circ,\quad\vec A'_2=(-13,0)\,\mathrm m$$ Adding $180^\circ$ gives $\phi\approx292.620^\circ$. The new pair is $(-13,0)$ m. Thus the condition $A_{y\prime}=0$ selects an undirected axis line; requiring $A_{x\prime}>0$ selects its sense.

(c) Interpret simplification geometrically.

$$A'^2=13^2+0^2=169\,\mathrm{m^2}=(-5)^2+12^2=A^2$$ The vector is now parallel to one axis, so its perpendicular projection vanishes. Its physical length remains 13 m. Coordinate choice can simplify a calculation by eliminating a component, but cannot turn a nonzero vector into the zero vector.