Why can't a line make obtuse angles with all the three coordinate axes?

Question

We were studying vectors and a question was that a line makes same angles with all the three axes,what is the value of direction cosines? The question is why can't a vector or line make an angle of 120° with all the coordinate axes?

Answer 1

I'm going to assume you're thinking of rays rather than lines, so that we're looking at the angles between a ray and the positive $x$, $y$, or $z$ axis.

Any ray can be described by a unit vector $\hat{\mathbf{n}} = (n_x,n_y,n_z)$ with $\hat{\mathbf{n}}\cdot\hat{\mathbf{n}} = 1$ that specifies the direction of the ray. The unit vectors for the positive coordinate axes are of course $\hat{\mathbf{x}}=(1,0,0)$, $\hat{\mathbf{y}}=(0,1,0)$, and $\hat{\mathbf{z}}=(0,0,1)$. The cosine of the angle between two rays is the dot product of their unit vectors. Thus, for our ray with unit vector $\hat{\mathbf{n}}$, we have $\cos\theta_x = n_x$, $\cos\theta_y = n_y$, and $\cos\theta_z = n_z$. However, from $\hat{\mathbf{n}}\cdot\hat{\mathbf{n}} = 1$ we then get $\cos^2\theta_x + \cos^2\theta_y + \cos^2\theta_z = 1$, which $\theta_x = \theta_y = \theta_z = 120^\circ$ does not satisfy. Therefore no ray can make a $120^\circ$ angle with all three positive coordinate axes.