91Ó°ÊÓ

The magnitude of a vector is essentially the length or size of the vector, and it indicates how strong or significant the vector is. For any vector \( \mathbf{v} = \langle a, b \rangle \), the formula for calculating the magnitude is:

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

This formula comes from the Pythagorean theorem, which relates the sides of a right triangle to its hypotenuse. Here, the vector components \( a \) and \( b \) are the triangle's legs, and the magnitude is the hypotenuse.

Substituting the vector in the exercise, \( \mathbf{v} = \left\langle -\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right\rangle \), into the formula, we find that:

• \( ||\mathbf{v}|| = \sqrt{ \left( -\frac{\sqrt{2}}{2} \right)^2 + \left( -\frac{\sqrt{2}}{2} \right)^2 } \)
• This simplifies to \( ||\mathbf{v}|| = \sqrt{ \frac{2}{4} + \frac{2}{4} } = \sqrt{ 1 } = 1 \)

Thus, the vector has a magnitude of 1, indicating it is a unit vector—meaning it has a length of one unit, which is useful in various applications.

The direction of a vector is the angle it forms with the positive x-axis. This angle, often represented by \( \theta \), shows where the vector points in the plane. To find the direction, we use the formula:

• \( \theta = \tan^{-1} \left( \frac{b}{a} \right) \)

Given our vector \( \mathbf{v} = \left\langle -\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right\rangle \), substitute the components to find the direction:

• \( \theta = \tan^{-1} \left( \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} \right) \)
• Since the ratio is 1, we initially find \( \theta = 45^\circ \)

However, since both components are negative, this places the vector in the third quadrant, meaning \( \theta \) is not \( 45^\circ \), but rather 45 degrees past 180 degrees. Therefore,\(
\)

• Actual direction \( \theta = 180^\circ + 45^\circ = 225^\circ \)

This shows how knowing the signs of the vector components helps us adjust the angle to fit the correct quadrant.

Understanding quadrants is crucial in determining the correct direction of a vector. The Cartesian plane is divided into four quadrants:

• First Quadrant: Both x and y are positive.
• Second Quadrant: x is negative, y is positive.
• Third Quadrant: Both x and y are negative.
• Fourth Quadrant: x is positive, y is negative.

Each quadrant corresponds to angles of a full circle, and knowing these helps in adjusting angles for vectors. Vectors with both components negative, like our example \( \left\langle -\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right\rangle \), are found in the third quadrant.

In trigonometry, the reference angle is the angle a vector makes with the x-axis, but to find the true angle from the positive x-axis, add 180 degrees for vectors in the third quadrant.

Understanding these quadrant-related adjustments is key to accurately describing the vector's direction.