Unit Vectors — Question 8

Question 8

The radial direction depends on position

Points P and Q have position vectors $(3\hat\imath+4\hat\jmath)$ m and $(-4\hat\imath+3\hat\jmath)$ m relative to O. At a nonzero position $\vec r$, define the outward radial unit vector as $\hat e_r=\vec r/|\vec r|$.

1. Find $\hat e_r$ at P and Q. Are they the same unit vector?

2. At P find the unit direction obtained by rotating $\hat e_r$ counterclockwise by $90^\circ$.

3. An outward 2 m displacement is prescribed at either point. Write its components at each point and explain why one fixed pair cannot describe both.

Question 8 - Solution

The radial direction depends on position

(a) Normalize at each location.

$$\boxed{\hat e_r(P)=(3,4)/5,\qquad\hat e_r(Q)=(-4,3)/5}$$ Both radii are 5 m. Hence $\hat e_r(P)=3\hat\imath/5+4\hat\jmath/5$ and $\hat e_r(Q)=-4\hat\imath/5+3\hat\jmath/5$. Both have unit length but different directions. A local radial direction changes as the position changes.

(b) Rotate components consistently.

$$\hat e_\theta(P)=(-4,3)/5,\quad\hat e_r(P)\cdot\hat e_\theta(P)=(-12+12)/25=0$$ A counterclockwise right-angle turn maps $(u_x,u_y)$ to $(-u_y,u_x)$. At P the requested direction is $-4\hat\imath/5+3\hat\jmath/5$, coincidentally equal to $\hat e_r(Q)$. It is tangent to the circle centered at O through P.

(c) Use the local direction in each prescription.

$$\vec d_P=2(3,4)/5=(1.2,1.6)\,\mathrm m,\quad\vec d_Q=(-1.6,1.2)\,\mathrm m$$ At P, the displacement is $(1.2\hat\imath+1.6\hat\jmath)$ m. At Q it is $(-1.6\hat\imath+1.2\hat\jmath)$ m. Equal outward lengths do not imply equal free vectors, because “outward refers to a different radial direction at each position.