Direction Cosines

Direction Cosines of a Vector

Consider a vector

\[ \mathbf{v} = x\hat{i} + y\hat{j} + z\hat{k} \]

with its initial point at the origin \( O(0,0,0) \) and terminal point at \( P(x, y, z) \). This vector makes angles \( \alpha, \beta, \gamma \) with the positive \( x \)-, \( y \)-, and \( z \)-axes, respectively.

To measure these angles, we consider right-angled triangles by constructing a cuboid where \( OP \) is its diagonal. This is done by drawing three planes parallel to the \( xy \)-, \( yz \)-, and \( zx \)-planes passing through \( P(x, y, z) \), as shown in the given figure.

• In triangle \( OAP \), we have:

  \[ \cos\alpha = \frac{OA}{OP} = \frac{x}{\sqrt{x^2 + y^2 + z^2}} \]

• In triangle \( OBP \), we have:

  \[ \cos\beta = \frac{OB}{OP} = \frac{y}{\sqrt{x^2 + y^2 + z^2}} \]

• In triangle \( OCP \), we have:

  \[ \cos\gamma = \frac{OC}{OP} = \frac{z}{\sqrt{x^2 + y^2 + z^2}} \]

Thus, the direction cosines of \( \mathbf{v} \) are:

\[ l = \cos\alpha = \frac{x}{|\mathbf{v}|}, \quad m = \cos\beta = \frac{y}{|\mathbf{v}|}, \quad n = \cos\gamma = \frac{z}{|\mathbf{v}|} \]

where

\[ |\mathbf{v}| = \sqrt{x^2 + y^2 + z^2}. \]

These cosines describe the orientation of the vector with respect to the coordinate axes.

• If \( \cos\alpha > 0 \), then \( \alpha \) is an acute angle with the \( x \)-axis.
• If \( \cos\alpha < 0 \), then \( \alpha \) is an obtuse angle with the \( x \)-axis.
• If \( \cos\alpha = 0 \), the vector is perpendicular to the \( x \)-axis.

Similar interpretations hold for \( \cos\beta \) and \( \cos\gamma \).

Also, rewriting \( \mathbf{v} \) using its magnitude and direction cosines:

\[ \mathbf{v} = x\hat{i} + y\hat{j} + z\hat{k} \]

\[ = \sqrt{x^2 + y^2 + z^2} \left( \frac{x}{\sqrt{x^2 + y^2 + z^2}} \hat{i} + \frac{y}{\sqrt{x^2 + y^2 + z^2}} \hat{j} + \frac{z}{\sqrt{x^2 + y^2 + z^2}} \hat{k} \right) \]

\[ = |\mathbf{v}| \left( \cos\alpha \hat{i} + \cos\beta \hat{j} + \cos\gamma \hat{k} \right). \]

Thus,

\[ \mathbf{u} = \cos\alpha \hat{i} + \cos\beta \hat{j} + \cos\gamma \hat{k} \]

is a unit vector along \( \mathbf{v} \), since its magnitude satisfies:

\[ |\mathbf{u}| = \sqrt{\cos^2\alpha + \cos^2\beta + \cos^2\gamma} = \sqrt{\frac{x^2}{x^2 + y^2 + z^2} + \frac{y^2}{x^2 + y^2 + z^2} + \frac{z^2}{x^2 + y^2 + z^2}} = 1. \]

This fundamental identity,

\[ \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1, \]

is always true for any nonzero vector in three-dimensional space.

Direction Ratios

Let us now define the concept of Direction Ratios in three-dimensional geometry.

Suppose there is a straight line in \(\mathbb{R}^3\). Since a straight line has no preferred direction, it is described up to orientation. That is, we are interested in the orientation of the line, not its sense (direction from one end to the other).

To capture the orientation of a line, one may consider any vector that is parallel to it. This vector is not unique. In fact, infinitely many vectors are parallel to a given line, differing only by scalar multiples. If \( \vec{v} \) is parallel to the line, then so is \( -\vec{v} \), \( 2\vec{v} \), \( \frac{1}{5}\vec{v} \), and so on.

For example, if \( \vec{v} = 2\hat{i} + \hat{j} - \hat{k} \) is parallel to a line, then so is \( 4\hat{i} + 2\hat{j} - 2\hat{k} \), since this is just \( 2\vec{v} \). These vectors are collinear—they lie along the same line in space and preserve the same orientation.

The key observation here is that to describe the orientation of the line, we do not need the exact vector. We only need the proportionality among its components. This leads us to the notion of direction ratios.

Definition:

If a line is parallel to a vector whose components are proportional to \( a, b, c \), then we say that the line has direction ratios \( (a, b, c) \).

Direction ratios are numbers that define the relative proportions of a vector's components in the \( x \)-, \( y \)-, and \( z \)-directions. They are free from scaling, i.e., the ratios \( (a, b, c) \) and \( (ka, kb, kc) \) represent the same direction ratios for any nonzero scalar \( k \). Also, direction ratios do not distinguish between opposite directions—\( (a, b, c) \) and \( (-a, -b, -c) \) describe the same orientation.

Notation:

Direction ratios are often written as an ordered triple:

\[ (a, b, c) \]

rather than using the ratio notation \( a : b : c \). This reflects that what matters is the proportional structure among the components, not their magnitude or sign individually.

Example:

If a line has direction ratios \( (2, 1, -1) \), then a vector parallel to it could be

\[ \vec{v} = 2\hat{i} + \hat{j} - \hat{k}. \]

Alternatively, one could take

\[ \vec{v} = 4\hat{i} + 2\hat{j} - 2\hat{k}, \quad \text{or} \quad \vec{v} = -2\hat{i} - \hat{j} + \hat{k}. \]

All of these define the same orientation of the line.

Direction ratios describe the orientation of a line in space by giving any triple of numbers proportional to the components of a vector parallel to the line. They are independent of magnitude and sense.

Direction ratios do not introduce any new idea that we don’t already get from a vector parallel to the line. They are just a simpler way to talk about the orientation of a line by looking at the ratio between the \( x \)-, \( y \)-, and \( z \)-components of a vector. Since any vector that is parallel to the line will do the job—no matter how long or short, or in which direction—it really comes down to what you find more convenient. You can choose to use direction ratios like \( (a, b, c) \), or you can use a full vector like \( a\hat{i} + b\hat{j} + c\hat{k} \). Both tell you the same thing about how the line is oriented in space.

Direction Ratios of a Line joining two Points

The direction ratios of a straight line joining two points

\( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \)
is given by the triple

\[ (x_1 - x_2,\ y_1 - y_2,\ z_1 - z_2). \]

This is because the vector \( \vec{BA} \) is parallel to the line passing through points \( A \) and \( B \), and any vector parallel to a line can be used to find its direction ratio.

We compute the vector \( \vec{BA} \) as follows:

\[ \vec{BA} = \vec{OA} - \vec{OB} = (x_1\hat{i} + y_1\hat{j} + z_1\hat{k}) - (x_2\hat{i} + y_2\hat{j} + z_2\hat{k}) \]

\[ \implies \vec{BA} = (x_1 - x_2)\hat{i} + (y_1 - y_2)\hat{j} + (z_1 - z_2)\hat{k}. \]

Hence, the components of \( \vec{BA} \) are the direction ratios of the line joining \( A \) and \( B \). That is, the direction ratio is:

\[ (x_1 - x_2,\ y_1 - y_2,\ z_1 - z_2). \]

Some Properties of Direction Ratios

Let \( a, b, c \) be the direction ratios of an undirected line in space. Then the following statements hold:

1. Any vector of the form

   \[ \vec{v} = a\hat{i} + b\hat{j} + c\hat{k} \]

   is parallel to the line. Conversely, if a vector \( \vec{v} \) is parallel to a line, then the components of \( \vec{v} \) form a valid set of direction ratios for that line.

2. Two lines are parallel if and only if their direction ratios are proportional. That is, if one line has direction ratios \( (a_1, b_1, c_1) \) and another has direction ratios \( (a_2, b_2, c_2) \), then the lines are parallel if and only if

   \[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}. \]

3. Two lines are perpendicular if and only if the dot product of their direction ratio vectors is zero. That is, lines with direction ratios \( (a_1, b_1, c_1) \) and \( (a_2, b_2, c_2) \) are perpendicular if and only if

   \[ a_1a_2 + b_1b_2 + c_1c_2 = 0. \]

4. Angle between two lines with direction ratios \( (a_1, b_1, c_1) \) and \( (a_2, b_2, c_2) \) is given by the cosine formula:

   \[ \cos\theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \cdot \sqrt{a_2^2 + b_2^2 + c_2^2}}. \]

5. Suppose the direction ratios of a line are \( (a, b, c) \). Then any vector of the form

   \[ \vec{v} = a\hat{i} + b\hat{j} + c\hat{k} \quad \text{or} \quad \vec{v} = -a\hat{i} - b\hat{j} - c\hat{k} \]

   is parallel to the line. These are the only two possibilities up to scaling, since a line has orientation but no preferred sense.

   The unit vectors in these two directions are obtained by dividing by the magnitude:

   \[ |\vec{v}| = \sqrt{a^2 + b^2 + c^2}, \]

   so the unit vectors parallel to the line are:

   \[ \mathbf{u}_1 = \frac{a}{\sqrt{a^2 + b^2 + c^2}}\hat{i} + \frac{b}{\sqrt{a^2 + b^2 + c^2}}\hat{j} + \frac{c}{\sqrt{a^2 + b^2 + c^2}}\hat{k}, \]

   and

   \[ \mathbf{u}_2 = -\frac{a}{\sqrt{a^2 + b^2 + c^2}}\hat{i} - \frac{b}{\sqrt{a^2 + b^2 + c^2}}\hat{j} - \frac{c}{\sqrt{a^2 + b^2 + c^2}}\hat{k}. \]

   Thus, the direction cosines of vectors parallel to the line are:

   \[ \left( \pm \frac{a}{\sqrt{a^2 + b^2 + c^2}},\ \pm \frac{b}{\sqrt{a^2 + b^2 + c^2}},\ \pm \frac{c}{\sqrt{a^2 + b^2 + c^2}} \right), \]

   where the signs must be taken consistently (either all \( + \) or all \( - \)). This reflects the fact that the line is undirected, but vectors have direction.