91Ó°ÊÓ

Understanding the magnitude of a vector is essential in vector mathematics. The magnitude, or length, of a vector represents how long the vector is in a geometric space. It's like finding the length of a segment in a line, but for vectors in a coordinate plane.
To calculate the magnitude of a vector with components \( \mathbf{a} = \langle a_1, a_2 \rangle \), you use the Pythagorean theorem. The formula is:

• \( \| \mathbf{a} \| = \sqrt{a_1^2 + a_2^2} \)

For the vector \( \mathbf{a} = \langle 0, -5 \rangle \), we identify \( a_1 = 0 \) and \( a_2 = -5 \). Plugging these into the formula gives us:

• \( \| \mathbf{a} \| = \sqrt{0^2 + (-5)^2} = \sqrt{25} = 5 \)

Thus, the magnitude of the vector is 5 units. This method works for any two-dimensional vector, helping to determine its length in the plane.

The direction angle of a vector is the angle it forms with the positive x-axis, measured counterclockwise. This angle helps understand the vector's orientation in space.
For a vector \( \mathbf{a} = \langle a_1, a_2 \rangle \), you can find this angle \( \theta \) using the tangent function:

• \( \tan(\theta) = \frac{a_2}{a_1} \)

In the case of \( \mathbf{a} = \langle 0, -5 \rangle \), since \( a_1 = 0 \), the direction points perpendicular to the x-axis.
The positive x-axis direction when the vector points downwards is:

• \( \theta = 270^\circ \)

This angle indicates the vector is directed downward along the negative y-axis.

A vector in two dimensions can be expressed with components, which are essentially the projections of the vector along the x- and y-axes.
For any vector \( \mathbf{a} \) expressed as \( \langle a_1, a_2 \rangle \):

• \( a_1 \) is the x-component. It shows how far the vector moves horizontally.
• \( a_2 \) is the y-component, indicating vertical movement.

For example, the vector \( \mathbf{a} = \langle 0, -5 \rangle \) has:

• x-component = 0, signifying no horizontal change;
• y-component = -5, meaning the vector moves 5 units in the negative y direction.

Understanding these components helps visualize and calculate other vector properties, such as direction and magnitude, easily.