Q.Find the direction cosines of a vector whose direction ratios are

Resolving in the plane. Let $\hat i,\hat j$ be unit vectors along the positive $x$- and $y$-axes. Every position vector in the plane is written uniquely as $\vec{OP}=x\hat i+y\hat j,\qquad |\vec{OP}|=\sqrt{x^2+y^2}.$ More generally, if $\vec a,\vec b$ are any two non-collinear vectors in a plane, every vector in that plane is a unique linear combination $\lambda\vec a+\mu\vec b$.

Resolving in space. With $\hat i,\hat j,\hat k$ along the positive $x,y,z$ axes, every position vector in space is uniquely $\vec{OP}=x\hat i+y\hat j+z\hat k,\qquad |\vec{OP}|=\sqrt{x^2+y^2+z^2}.$ Three non-coplanar vectors $\vec a,\vec b,\vec c$ likewise span all of space uniquely: any vector is $\lambda\vec a+\mu\vec b+\nu\vec c$. The vector joining $(x_1,y_1,z_1)$ to $(x_2,y_2,z_2)$ has components $(x_2-x_1)\hat i+(y_2-y_1)\hat j+(z_2-z_1)\hat k$.

Matrix form. $\vec A=a_1\hat i+a_2\hat j+a_3\hat k$ can be written as the column $\begin{pmatrix}a_1\\a_2\\a_3\end{pmatrix}$; addition and scalar multiplication of vectors then match ordinary matrix addition and scalar multiplication component-by-component.

Direction cosines and direction ratios. For a point $P(x,y,z)$ at distance $r=\sqrt{x^2+y^2+z^2}$ from the origin, let $\alpha,\beta,\gamma$ be the angles $\vec{OP}$ makes with the positive $x,y,z$ axes (the direction angles). Then $\cos\alpha=\frac{x}{r},\quad \cos\beta=\frac{y}{r},\quad \cos\gamma=\frac{z}{r}$ are the direction cosines of $\vec{OP}$, usually written $(l,m,n)$. Any triple proportional to $(l,m,n)$ — such as $(x,y,z)$ itself — is a set of direction ratios; unlike direction cosines, direction ratios are not unique.

Key facts:

• $l^2+m^2+n^2=1$ always (equivalently $\sin^2\alpha+\sin^2\beta+\sin^2\gamma=2$).
• Conversely, three numbers are the direction cosines of some vector iff the sum of their squares is $1$ — this is exactly what you check to verify a given triple.
• Any unit vector can be written as $\cos\alpha\,\hat i+\cos\beta\,\hat j+\cos\gamma\,\hat k$.
• Knowing only direction ratios or only direction cosines does not pin down a vector — you also need its magnitude.