91Ó°ÊÓ

When you have multiple vectors and you want to combine them into a single vector, the outcome is known as a resultant vector. To find the resultant vector, you simply perform vector addition. This means adding the corresponding components of each vector together.

For example, if you have vectors \( \mathbf{a} = [a_x, a_y, a_z] \) and \( \mathbf{b} = [b_x, b_y, b_z] \), the resultant vector \( \mathbf{R} \) would be:

• \( R_x = a_x + b_x \)
• \( R_y = a_y + b_y \)
• \( R_z = a_z + b_z \)

For the exercise, the vectors \( \mathbf{p} = [-1, -3, -5] \), \( \mathbf{q} = [6, 4, 2] \), and \( \mathbf{u} = [-5, -1, 3] \) were added to provide the resultant vector \( \mathbf{R} = [0, 0, 0] \). As shown, each component addition resulted in zero, meaning these vectors neutralized each other perfectly.

The magnitude of a vector represents its length or size, and is always a non-negative number. For any vector \( \mathbf{v} = [v_x, v_y, v_z] \), the magnitude \( |\mathbf{v}| \) is calculated using the Pythagorean theorem extension in three dimensions:

\[|\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}\]

This formula gives you insight into how "long" the vector is, regardless of its direction in space. In the exercise, the resultant vector was \([0, 0, 0]\), making its magnitude calculated as \( \sqrt{0^2 + 0^2 + 0^2} = 0 \).

A magnitude of zero implies that the vector is just a point, having no actual "distance" from the origin. This occurs when vectors perfectly counter each other out.

Vectors are not just about direction; they also have magnitude, split across dimensions. Each vector can be broken down into components, which represent its influence along each axis. For a three-dimensional vector, these components are along the x, y, and z axes.

Consider a vector \( \mathbf{v} = [v_x, v_y, v_z] \), each component represents how far and in what direction the vector stretches along that axis:

• \( v_x \) is the influence in the horizontal x-axis direction.
• \( v_y \) is the influence in the vertical y-axis direction.
• \( v_z \) is the influence along the depth z-axis.

In the exercise, each given vector had clear components that, once added together with the components of the other vectors, resulted in the vector \( \mathbf{R} = [0, 0, 0] \). This demonstrates that vector addition relies heavily on understanding and correctly working with vector components.