Question

Example 38. A point P lies in the x-y plane. Its position
can be specified by its x, y coordinates or by a radially
directed vector r = (xi + y j ), making an angle theta with the
x-axis. Find a vector i(r), of unit magnitude in the direction of
vector r and a vector i (theta), of unit magnitude normal to the
vector î(r) and lying in the x-y plane.
[NCERT]​

Answer 1

Given: Vector r = x(i cap) + y(j cap), making an angle theta with the  x-axis

To find: A vector i(r), of unit magnitude in the direction of  vector r and a vector i (theta).

Solution:

• Now we have given the vector r = x(i cap) + y(j cap), so unit vector in vector r direction is:

            i = vector r / r(cap)

              = x(i cap) + y(j cap) / r

              = x/r i(cap) + y/r j(cap)

• Now r makes angle Ф with x axis as given in the question, so:

            cos Ф = x/r and sin Ф = y /r

• So, i = cos Ф i(cap) + sin Ф j(cap)

• Let i(Ф) = a i(cap) + b j(cap)

• But i(r) and i(Ф) are perpendicular, so:

             i(r) . i(Ф) = 0

             { cos Ф i(cap) + sin Ф j(cap) } .  { a i(cap) + b j(cap) } = 0

• Solving this, we get:

            a cosФ + b sin Ф = 0

            a = -b sin Ф / cos Ф

• As we know that i(Ф) is a unit vector, so a² + b² = 1

            a = - sinФ and b = cosФ

• So, i(Ф) =  - sinФ i (cap) +  cosФ j (cap)

Answer:

         So the vector i(Ф) =  - sinФ i (cap) +  cosФ j (cap)

Answer 2

Explanation:

THIS ANSWER IS AS PER NCERT TEXT BOOK