All Solutions
Page 125: Section Review
When for example we go from home to school and back the total distance is double the distance from home to school but the displacement in that case is zero because we are back to the starting point.
$$
begin{align*}
mathbf{K}&=-4 \
mathbf{L}&=6 \
mathbf{L-K}&=? \
\
mathbf{L-K}&=mathbf{L}+mathbf{-K} \
&=6+-(-4) \
&=10
end{align*}
$$
mathbf{L-K}=10
$$
The figure shows a graphical representation of the component form.
$$
begin{align*}
mathbf{M}&=5 \
theta &= 37text{textdegree} \
end{align*}
$$
Unknown:
$$
begin{align*}
mathbf{M_x}&=? \
mathbf{M_y}&=? \
\
mathbf{M_x}&=mathbf{M}cdot cos(theta) \
&=5cdot cos(37text{textdegree}) \
&=boxed{3.993 } \
\
mathbf{M_y}&=mathbf{M}cdot sin(theta) \
&=5cdot sin(37text{textdegree}) \
&=boxed{3.009 }
end{align*}
$$
begin{align*}
mathbf{M_x}&=3.993 \
mathbf{M_y}&=3.009
end{align*}
$$
$$
begin{align*}
mathbf{K}&=-4hat{x} \
mathbf{L}&=6hat{x} \
mathbf{M}&=3.993hat{x}+3.009hat{y} \
mathbf{R}&=? \
\
mathbf{R_x}&=mathbf{K}+mathbf{L}+mathbf{M}hat{x} \
&=-4+6+3.993 \
&=5.993hat{x} \
\
mathbf{R_y}&=mathbf{M}hat{y} \
&=3.009hat{y} \
\
R&=sqrt{mathbf{R^2_x}+mathbf{R^2_y}} \
&=sqrt{5.993^2+3.009^2} \
&=6.706
end{align*}
$$
$$
R=6.706
$$
$R_x = M_x + L_x – K_x = 4.0 + 6.0 – 4.0 = 6.0$
In the y direction:
$R_y = M_y + L_y + K_y = 3.0 + 0.0 + 0.0 = 3.0$
Thus the sum of the vectors is:
$$
R = sqrt{R_x^2 + R_y^2} = sqrt{(6.0)^2 + (3.0)^2} = 6.7
$$
When adding two vectors, we get the smallest sum when they are in the same direction and opposite orientation, but because they are of different magnitudes, that sum will never be zero.
When adding three vectors it is possible to get the sum of zeros, they just have to form a triangle.
