Vector: Direction and Ratios
Vector: Direction Cosines
Before discussing the directional cosines of a vector, let us discuss the position vector. Just like the name suggests, a position vector indicates the position of any point relative with respect to any reference origin.
Consider any arbitrary point in three-dimensional space having the coordinates (x,y,z) as concerning the origin O (0,0,0).
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The position vector in the above figure is given as |PQ| = r = √(x - 0)2 + (y - 0)2 + (z - 0)2
|PQ| = r = √x2 + y2 + z2
Direction Cosines
Consider the following figure that represents a vector P in space with variable O being the reference origin of the vector P. Let the position vector make a positive angle (anticlockwise direction) of α, β, and γ with the positive x, y, and z-axis respectively. These angles are known as direction angles. When we take the cosine of these angles, we can find out the direction cosines. Taking direction cosines makes it easy to represent the direction of a vector in terms of angles for reference.
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\[cos \alpha = \frac{x}{|\overline{r}|}\]
\[cos \beta = \frac{y}{|\overline{r}|}\]
\[cos \gamma = \frac{z}{|\overline{r}|}\]
This is how the cosines of the directions of a vector are represented mathematically.
\[cos \alpha = \frac{x}{\sqrt{x^{2} + y^{2} + z^{2}}}\]
\[cos \beta = \frac{y}{\sqrt{x^{2} + y^{2} + z^{2}}}\]
\[cos \gamma = \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}\]
Direction Ratios
The product of the magnitude of any given vector can be represented with point P, and the cosines of direction on the three axes, i.e.
a = lr
b = mr
c = nr
Where,
l = direction of the cosine on the axis X.
m = direction of the cosine on the axis Y.
n = direction of the cosine on the axis Z.
This helps to understand that lr, mr, and nr are in proportion to direction cosines. Hence, they are called direction ratios and are represented by the variables a, b and c.
Where the axes l, m, n represent the respective direction cosines of any given vector on the axes X, Y, Z respectively. We can see that lr, mr, nr are in proportion to the direction cosines and these are called the direction ratios and they are denoted by a, b, c.
\[\frac{l}{a} = \frac{1}{\sqrt{x^{2} + y^{2} + z^{2}}}\]
\[\frac{m}{b} = \frac{1}{\sqrt{x^{2} + y^{2} + z^{2}}}\]
\[\frac{n}{c} = \frac{1}{\sqrt{x^{2} + y^{2} + z^{2}}}\]
From the above theory, we have learned about direction ratios and direction cosines. Let us apply this knowledge and solve some problems.
Solved Examples
1. Consider the point A(x, y, z) having coordinates (3, 4, 5). Find the direction ratios and the direction cosines with the origin point being O(0, 0, 0).
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We have learned that,
\[cos \alpha = \frac{x}{|\overline{r}|}\]
\[cos \beta = \frac{y}{|\overline{r}|}\]
\[cos \gamma = \frac{z}{|\overline{r}|}\]
\[\Rightarrow \sqrt{x^{2} + y^{2} + z^{2}} = \sqrt{3^{2} + 4^{2} + 5^{2}}\]
\[\Rightarrow \sqrt{x^{2} + y^{2} + z^{2}} = \sqrt{50} = 5\sqrt{2}\]
Hence, we can conclude that:
\[l = cos \alpha = \frac{3}{5\sqrt{2}}\]
\[m = cos \beta = \frac{4}{5\sqrt{2}}\]
\[n = cos \gamma = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}}\]
The direction ratio of the given point A(x, y, z) will be 3:4:5.
⇒ x = 3
⇒ y = 4
⇒ z = 5
FAQs on How to Find Direction Ratios and Direction Cosines?
1. What are direction cosines (DCs) in three-dimensional geometry?
Direction cosines (DCs) of a line are the cosines of the angles that the line makes with the positive directions of the x, y, and z-axes. If a line makes angles α, β, and γ with the x, y, and z-axes respectively, its direction cosines are denoted by l, m, and n, where:
l = cos α
m = cos β
n = cos γ
These values define the orientation of the line in space. A fundamental property is that the sum of their squares is always 1, i.e., l² + m² + n² = 1.
2. What are direction ratios (DRs) and how do they relate to direction cosines?
Direction ratios (DRs) are any three numbers (let's call them a, b, c) that are proportional to the direction cosines (l, m, n) of a line. This means there exists a non-zero constant k such that:
a = kl, b = km, and c = kn
While a line has a unique set of direction cosines (up to sign), it can have infinitely many sets of direction ratios. You can find the direction cosines from direction ratios using the formula provided in our detailed explanations.
3. How do you find the direction ratios of a line segment joining two points?
To find the direction ratios of a line segment joining two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂), you simply subtract the coordinates of the initial point from the final point. The direction ratios (a, b, c) are given by:
a = x₂ – x₁
b = y₂ – y₁
c = z₂ – z₁
For example, the direction ratios of the line joining points (1, 2, 3) and (4, 6, 8) are (4-1, 6-2, 8-3), which simplifies to (3, 4, 5). More details can be found in our NCERT Solutions for Class 12 Maths Chapter 11.
4. How can you calculate the direction cosines of a line if you know its direction ratios?
If the direction ratios of a line are a, b, and c, you can find its direction cosines (l, m, n) by dividing each direction ratio by the square root of the sum of the squares of the direction ratios. The formulas are:
l = ± a / √(a² + b² + c²)
m = ± b / √(a² + b² + c²)
n = ± c / √(a² + b² + c²)
The ± sign indicates that a directed line can have two sets of direction cosines (l, m, n) and (-l, -m, -n), corresponding to the two opposite directions of the line.
5. What is the fundamental difference between direction ratios and direction cosines?
The fundamental difference lies in their uniqueness and magnitude.
Uniqueness: A line in space has only one set of direction cosines (or its negative, representing the opposite direction). However, it has an infinite number of sets of direction ratios, as any non-zero multiple of a set of DRs is also a valid set of DRs.
Magnitude: Direction cosines are geometrically defined as the cosines of angles and are constrained by the identity l² + m² + n² = 1. Direction ratios are just proportional numbers and have no such constraint on their magnitude.
6. Why is the sum of the squares of direction cosines (l² + m² + n²) always equal to 1?
This property arises directly from the geometric definition in a 3D coordinate system. Consider a vector r from the origin to a point P(x, y, z). The magnitude of this vector is |r| = √(x² + y² + z²). The direction cosines are l = x/|r|, m = y/|r|, and n = z/|r|. If we square and add them:
l² + m² + n² = (x/|r|)² + (y/|r|)² + (z/|r|)² = (x² + y² + z²) / |r|²
Since |r|² = x² + y² + z², the expression simplifies to 1. Geometrically, this signifies that the direction cosines are the components of a unit vector in the direction of the line.
7. How are direction ratios used to determine if two lines are parallel or perpendicular?
Direction ratios are crucial for analysing the relationship between two lines. Let two lines have direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂).
For Parallel Lines: Two lines are parallel if their direction ratios are proportional. This means there is a constant k such that: a₁/a₂ = b₁/b₂ = c₁/c₂ = k.
For Perpendicular Lines: Two lines are perpendicular if the sum of the products of their corresponding direction ratios is zero. The condition is: a₁a₂ + b₁b₂ + c₁c₂ = 0. This is derived from the dot product of their direction vectors being zero.
8. What are the direction cosines of the x-axis, y-axis, and z-axis?
Understanding the direction cosines of the coordinate axes is a key concept.
x-axis: The x-axis makes angles of 0° with the x-axis, 90° with the y-axis, and 90° with the z-axis. Therefore, its DCs are (cos 0°, cos 90°, cos 90°) = (1, 0, 0).
y-axis: The y-axis makes angles of 90°, 0°, and 90° with the x, y, and z-axes respectively. Its DCs are (cos 90°, cos 0°, cos 90°) = (0, 1, 0).
z-axis: The z-axis makes angles of 90°, 90°, and 0° with the x, y, and z-axes respectively. Its DCs are (cos 90°, cos 90°, cos 0°) = (0, 0, 1).
