I need help!!! about vector rectangular, cylindrical ,and sphrical coordinates problem!! Help me please?

Question

(Electromagnetism)

Can someone help me with this problem?? cuz i need it up till tommorow

At point P(-3,-4,5), express that vector that extends from P to Q(2,0,-1)in:

(a) rectangular coordinates;

(b) cylindrical coordinates ;

(c) spherical coordinates.

(d) Show that each of these vectors has the same magnitude

Please help me .... thnk you and God Bless!!

Answer 1

Let, Unit Vectors :

Rectangular  coordinates : (a_x), (a_y), (a_z).

Cylindrical coordinates : (a_ro), (a_phi), (az)

Spherical Coordinates : (a_r), (a_theta), (a_phi)

(a)

The vector from P to Q in rectangular coordinates is :

PQ = {2-(-3)}(a_x) + {0-(-4)}(a_y) + (-1-5)(a_z)

= 5(a_x) + 4(a_y) - 6(a_z)

Magnitude = sqrt(5^2 + 4^2 + 6^2)

(b)

PQ_ro = {5(a_x) + 4(a_y) - 6(a_z)} . (a_ro)

= 5cos(phi) - 8sin(phi)

Simiplarly, PQ_phi = PQ . (a_phi) = -10sin(phi) - 8cos(phi)

PQ_z = -6

ro = sqrt(5^2 + 4^2)

phi = tan(inverse)(4/5)

put the value of phi to find PQ_ro, PQ_phi, PQ_z

PQ in cylindrical coordinate is PQ_ro + PQ_phi + PQ_z

(c)

PQ_r = PQ . (a_r)

PQ_theta = PQ . (a_theta)

PQ_phi = PQ . (a_phi)

Hint :

a_x . a_r = sin(theta)cos(phi)

a_x . a_theta = cos(theta)cos(phi)

a_x . a_phi = -sin(phi)

a_y . a_r = sin(theta)sin(phi)

a_y . a_theta = cos(theta)sin(phi)

a_y . a_phi = cos(theta)

a_z . a_r = cos(theta)

a_z . a_theta = -sin(theta)

a_z . a_phi = 0

find r = sqrt(5^2 + 4^2 + 6^2)

theta = cos(inverse) (-6/r)

phi = tan(inverse) (4/5)

Finally, PQ in spherical coordinates :

PQ = (PQ_r)(a_r) + (PQ_theta)(a_theta) + (PQ_phi)(a_phi)

Find the magnitudes as you did earlier.

All the best.