We are given that,

is inclined at 60° to and at 45° to .

We need to find the position vector of point A in space.

That is,

Position vector of A =

So, we need to find .

We know that, space contains three axes, namely, X, Y and Z.

Let OA be inclined at angle α with OZ.

And we know that direction cosines are associated by the relation,

l² + m² + n² = 1 …(i)

And here, direction cosines are cosines of the angles inclined by on , and . So,

l = cos 60°

m = cos 45°

n = cos α

Substituting these values of l, m and n in equation (i), we get

(cos 60°)² + (cos 45°)² + (cos α)² = 1

We know the values of cos 60° and cos 45°.

That is,

We get,

So is given as,

…(ii)

We have,

Putting these values of l, m and n in equation (ii), we get

Also, put .

Thus, position vector of point A in space is .