Vectors - Direction Cosines and Direction Ratios

Direction Cosines (DCs) are the cosines of the angles $\alpha$, $\beta$, and $\gamma$ that a vector makes with the positive directions of the $x$, $y$, and $z$ axes respectively. Visually, if you imagine a vector $\vec{r}$ originating from the origin $O$ to a point $P$, $\alpha$ is the angle between the vector and the $x$-axis, $\beta$ with the $y$-axis, and $\gamma$ with the $z$-axis.

Direction Cosines are typically denoted by the letters $l$, $m$, and $n$, where $l = \cos \alpha$, $m = \cos \beta$, and $n = \cos \gamma$. A key geometric property is that these values represent the coordinates of a point on a unit sphere centered at the origin, meaning the magnitude of the vector $(l, m, n)$ is always $1$.

Direction Ratios (DRs) are any three numbers $a, b, c$ that are proportional to the direction cosines $l, m, n$. While a vector has a unique set of direction cosines, it can have infinitely many sets of direction ratios. Visually, any vector parallel to the original vector will share the same direction ratios, differing only by a scalar multiple $k$.

The sum of the squares of the direction cosines of a line is always equal to $1$, expressed as $l^2 + m^2 + n^2 = 1$. This implies that $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$. This identity is crucial for finding a missing angle or cosine value when the other two are known.

If a vector is given as $\vec{r} = a\hat{i} + b\hat{j} + c\hat{k}$, the components $a, b, c$ are the direction ratios of the vector. To visualize this, these components represent the steps taken along the three primary axes to reach the terminal point of the vector from its starting point.

The direction cosines of a line segment joining two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ are found by first determining the direction ratios $(x_2 - x_1), (y_2 - y_1), (z_2 - z_1)$, and then dividing each by the distance $PQ$. Visually, this normalizes the displacement vector between the two points to a unit length.

Two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ are parallel if $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$. They are perpendicular if the sum of the products of their corresponding ratios is zero, i.e., $a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$. This dot-product relationship is fundamental in 3D geometry.