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
