91Ó°ÊÓ

Vector magnitude refers to the length or size of a vector. It is calculated using the components of the vector itself. For example, in three-dimensional space, a vector \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \) has a magnitude given by:

• \( |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} \)

The magnitude provides a scalar value which represents how long the vector is, independent of its direction.
In our example, for \( \mathbf{v} = \mathbf{i} + \mathbf{j} - \mathbf{k} \), the magnitude is \( \sqrt{1 + 1 + 1} = \sqrt{3} \). Similarly, for another vector \( \mathbf{v} = 2\mathbf{i} - 2\mathbf{j} + \mathbf{k} \), the magnitude is \( \sqrt{4 + 4 + 1} = 3 \). These magnitudes help in understanding and visualizing the size of vectors in the solutions.

Direction angles are the angles that a vector makes with the positive x, y, and z-axes. These angles, denoted by \( \alpha, \beta, \text{and} \gamma \), help in expressing the direction of the vector. To find these angles, one uses the direction cosines. The direction angles satisfy the equations:

• \( \alpha = \cos^{-1}(l) \)
• \( \beta = \cos^{-1}(m) \)
• \( \gamma = \cos^{-1}(n) \)

For the vector \( \mathbf{v} = \mathbf{i} + \mathbf{j} - \mathbf{k} \), the direction angles are approximately \( 55^\circ, 55^\circ, \text{and} 125^\circ \) respectively. Similarly, for \( \mathbf{v} = 2\mathbf{i} - 2\mathbf{j} + \mathbf{k} \), these angles are approximately \( 48^\circ, 132^\circ, \text{and} 71^\circ \).
This approximation helps provide a more intuitive sense of the vector's orientation in space.

Equation (5) is fundamental in the understanding of direction cosines of a vector. It states that the sum of the squares of the direction cosines should equal one. This is mathematically represented as:

• \( l^2 + m^2 + n^2 = 1 \)

Here, \( l, m, \text{and} n \) are the direction cosines, and they are calculated from the vector components divided by its magnitude.
For instance, for the vector \( \mathbf{v} = \mathbf{i} + \mathbf{j} - \mathbf{k} \), the calculation is \( \left( \frac{1}{\sqrt{3}} \right)^2 + \left( \frac{1}{\sqrt{3}} \right)^2 + \left( \frac{-1}{\sqrt{3}} \right)^2 = 1 \). Similarly, for \( \mathbf{v} = 2\mathbf{i} - 2\mathbf{j} + \mathbf{k} \), verify \( \left( \frac{2}{3} \right)^2 + \left( \frac{-2}{3} \right)^2 + \left( \frac{1}{3} \right)^2 = 1 \).
This equation confirms that direction cosines are calculated accurately, ensuring the representation of vector orientation is correct.