Linear Algebra, Theory And Applications, 2012a

2.7. EXERCISES 71
this will suffice to explain the above question. Then the acceleration observed by a person
on the earth relative to the apparently fixed vectors, i, k, j, is
a B = −a (r B )(xi+yj+zk) − 2ω [−y ′ cos (φ) i+(x ′ cos (φ)+z ′ sin (φ)) j− (y ′ sin (φ) k)]
Therefore, one obtains some differential equations from a B = x ′′ i + y ′′ j + z ′′ k by matching
the components. These are
x ′′ + a (r B ) x = 2ωy ′ cos φ
y ′′ + a (r B ) y = −2ωx ′ cos φ − 2ωz ′ sin (φ)
z ′′ + a (r B ) z = 2ωy ′ sin φ
Now remember, the vectors, i, j, k are fixed relative to the earth and so are constant vectors.
Therefore, from the properties of the determinant and the above differential equations,
′
(r ′ B × r B ) ′ i j k
i j k
=
x ′ y ′ z ′
=
x ′′ y ′′ z ′′
∣ x y z ∣ ∣ x y z ∣
i j k
=
−a (r B ) x +2ωy ′ cos φ −a (r B ) y − 2ωx ′ cos φ − 2ωz ′ sin (φ) −a (r B ) z +2ωy ′ sin φ
∣
x y z ∣
Then the k th component of this cross product equals
ω cos (φ) ( y 2 + x 2) ′
+2ωxz ′ sin (φ) .
The first term will be negative because it is assumed p (t) is the location of low pressure
causing y 2 +x 2 to be a decreasing function. If it is assumed there is not a substantial motion
in the k direction, so that z is fairly constant and the last term can be neglected, then the
k th component of (r ′ B × r B) ′ is negative provided φ ∈ ( 0, π 2
)
and positive if φ ∈
( π
2 ,π) .
Beginning with a point at rest, this implies r ′ B × r B = 0 initially and then the above implies
its k th component is negative in the upper hemisphere when φπ/2. Using the right hand and the geometric definition of the
cross product, this shows clockwise rotation in the lower hemisphere and counter clockwise
rotation in the upper hemisphere.
Note also that as φ gets close to π/2 near the equator, the above reasoning tends to
break down because cos (φ) becomes close to zero. Therefore, the motion towards the low
pressure has to be more pronounced in comparison with the motion in the k direction in
order to draw this conclusion.
2.7 Exercises
1. Show the map T : R n → R m defined by T (x) =Ax where A is an m × n matrix and
x is an m × 1 column vector is a linear transformation.
2. Find the matrix for the linear transformation which rotates every vector in R 2 through
an angle of π/3.
3. Find the matrix for the linear transformation which rotates every vector in R 2 through
an angle of π/4.
4. Find the matrix for the linear transformation which rotates every vector in R 2 through
an angle of −π/3.