1. Magnitude of a Vector
The magnitude (or length) of a vector tells how long the vector is. It is always a positive number.
For a vector in two dimensions, \(\vec{v} = \langle x, y \rangle\), the magnitude is:
\[ |\vec{v}| = \sqrt{x^2 + y^2} \]
For a vector in three dimensions, \(\vec{v} = \langle x, y, z \rangle\), the formula becomes:
\[ |\vec{v}| = \sqrt{x^2 + y^2 + z^2} \]
1.1. Example of Computing Magnitude
For the vector \(\vec{v} = \langle 3, 4 \rangle\):
\[ |\vec{v}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 \]
This is the familiar 3–4–5 right triangle.
2. Direction and Unit Vector
The direction of a vector describes the way it points. A convenient way to express direction is by using a unit vector.
A unit vector has magnitude 1 and points in the same direction as the original vector.
To convert a vector \(\vec{v}\) into a unit vector, divide it by its magnitude:
\[ \hat{v} = \dfrac{\vec{v}}{|\vec{v}|} \]
This gives a vector that points the same way but has length 1.
2.1. Example of a Unit Vector
For \(\vec{v} = \langle 6, 8 \rangle\):
Magnitude:
\[ |\vec{v}| = \sqrt{6^2 + 8^2} = 10 \]
Unit vector:
\[ \hat{v} = \left\langle \dfrac{6}{10}, \dfrac{8}{10} \right\rangle = \langle 0.6, 0.8 \rangle \]
3. Direction Ratios and Direction Cosines
Direction ratios (D.R.) and direction cosines (D.C.) help describe the direction of a vector in three-dimensional space.
If a vector is \(\vec{v} = \langle a, b, c \rangle\), then:
- The numbers \(a, b, c\) are its direction ratios.
- The direction cosines are the cosines of the angles the vector makes with the positive x-, y-, and z-axes.
If the vector makes angles \(\alpha, \beta, \gamma\) with the x-, y-, z-axes respectively, then:
\[ \cos \alpha = \dfrac{a}{\sqrt{a^2 + b^2 + c^2}}, \,\, \cos \beta = \dfrac{b}{\sqrt{a^2 + b^2 + c^2}}, \,\, \cos \gamma = \dfrac{c}{\sqrt{a^2 + b^2 + c^2}} \]
3.1. Example of Direction Ratios and Cosines
For the vector \(\vec{v} = \langle 2, 3, 6 \rangle\):
Magnitude:
\[ |\vec{v}| = \sqrt{2^2 + 3^2 + 6^2} = \sqrt{49} = 7 \]
Direction cosines:
\[ \cos \alpha = \dfrac{2}{7}, \quad \cos \beta = \dfrac{3}{7}, \quad \cos \gamma = \dfrac{6}{7} \]
4. Summary of Direction and Magnitude
- Magnitude shows how long the vector is.
- Direction shows which way it points.
- A unit vector gives the pure direction.
- Direction ratios and cosines help describe direction in 3D.
Together, magnitude and direction fully describe any vector.