If the cosine angle is given for only X and Y axis, but missing (not mentioned) for the Z axis. How many lines can be represented by the two given direction cosines. The text book say two, one forming an acute angle with Z axis and another forming obtuse.
I would have thought the answer to be infinitely many lines. I know I am missing something and I struggle to visualize the angle with the axis (more easy to think of angle with a plane).
In my opinion if we take a point $(x_1,y_1,z_1)$ and say a line passing through the origin and this point makes angle $\alpha$ with X axis, $\beta$ with Y axis. Then there also exists a plane inclined at same angles ($\alpha$ with X axis, $\beta$ with Y axis) containing this line. And since the plane has infinitely many lines there should be infinite lines with given two direction cosines.
I understand that the third direction cosine does uniquely identify the line in space. Question is only when only two are given.
And I also understand algebraically solving $cos^2\alpha + cos^2\beta + cos^2\gamma = 1$ will yield two values for $\gamma$. Its just that I am not able to visualize this given my reasoning above.
You seem to think that if a plane makes an angle $\theta$ with a coordinate axis then every vector in that plane makes the same angle.
Consider the plane $y=x$. It makes a $45^{\circ}$ angle with the $x$ axis but it contains the $z$ axis which is at an angle of $90^{\circ}$ with the $x$ axis.