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Direction of a Vector

The direction of a vector is expressed as the angle \( \alpha \) formed between the vector and the positive x-axis in a Cartesian coordinate system.
vector direction

To calculate this angle, we use a fundamental geometric principle: the tangent of an angle in a right triangle.

A vector \( \vec{v} \) can be represented by its components \( x \) and \( y \).

On a Cartesian plane, the vector can be visualized as the diagonal of a rectangle whose sides are the absolute values of \( |x| \) and \( |y| \), ensuring all lengths are positive.

the vector

The relationship between the angle \( \alpha \), the opposite side (\( y \)), and the adjacent side (\( x \)) is expressed by the formula:

$$ \tan \alpha = \frac{|y|}{|x|} $$

To find \( \alpha \), we calculate the inverse tangent (arctangent) of the ratio, which converts the sides' proportions into an angle measured in radians or degrees:

$$ \alpha = \arctan \left( \frac{|y|}{|x|} \right) $$

This formula reveals that the vector's direction depends solely on the ratio of its components.

Note that \( \alpha \) will always be an acute angle (between \( 0^\circ \) and \( 90^\circ \)) when both components are positive.

However, as we will see in the examples, the signs of \( x \) and \( y \) determine the vector's quadrant on the plane, which affects the calculation of \( \alpha \).

If one or both components are negative, we need to adjust the angle obtained from the arctangent based on the vector's quadrant.

Understanding a vector's direction not only clarifies "where" it points but is also critical for accurately describing phenomena involving motion, forces, or other vector quantities. Miscalculating the angle can lead to significant errors, such as misinterpreting the motion of an object or a natural event.

Practical Example

Suppose we have a vector with components \( x = 3 \) and \( y = 4 \). Here's how to calculate its direction:

First, compute the ratio \( \frac{|y|}{|x|} \):

$$ \frac{|y|}{|x|} = \frac{4}{3} \approx 1.333 $$

Next, apply the arctangent function:

$$ \alpha = \arctan(1.333) $$

Using a calculator, we find:

$$ \alpha \approx 53.13^\circ $$

Thus, the vector's direction is approximately 53.13 degrees relative to the x-axis.

vector angle

Calculating a vector's direction using tangent and arctangent is a straightforward and reliable method, demonstrating how geometry and trigonometry combine to explain complex concepts.

Example 2

Let’s consider a different scenario.

If the vector's components are \( x = -3 \) and \( y = 4 \), the calculation changes because the signs of the components determine the quadrant in which the vector lies. Here's the process:

As before, we start by calculating the ratio of the absolute values:

$$ \frac{|y|}{|x|} = \frac{4}{3} \approx 1.333 $$

The acute angle \( \alpha \), relative to the positive x-axis, is:

$$ \alpha = \arctan\left(\frac{|y|}{|x|}\right) = \arctan(1.333) $$

Using a calculator, we find:

$$ \alpha \approx 53.13^\circ $$

Since \( x = -3 \) (negative) and \( y = 4 \) (positive), the vector lies in the second quadrant of the plane.

In this quadrant, the direction of the vector is measured counterclockwise from the positive x-axis and is given by:

$$ \theta = 180^\circ - \alpha $$

Subtract the acute angle \( \alpha \) from \( 180^\circ \):

$$ \theta = 180^\circ - 53.13^\circ \approx 126.87^\circ $$

Thus, the vector's direction, considering its positive and negative components, is:

$$ \theta \approx 126.87^\circ $$

When plotted on a Cartesian plane with \( x = -3 \) and \( y = 4 \), the vector points toward the second quadrant.

vector direction

The value \( 126.87^\circ \) represents the angle relative to the positive x-axis, following mathematical conventions.

This highlights the importance of considering the quadrant when determining the direction of a vector.

Example 3

If the vector's components are \( x = -3 \) and \( y = -4 \), the calculation differs again since the vector is now in the third quadrant.

Here’s the step-by-step approach:

First, calculate the ratio using the absolute values of the components:

$$ \frac{|y|}{|x|} = \frac{4}{3} \approx 1.333 $$

The acute angle \( \alpha \), relative to the positive x-axis, is:

$$ \alpha = \arctan\left(\frac{|y|}{|x|}\right) = \arctan(1.333) $$

Using a calculator, we find:

$$ \alpha \approx 53.13^\circ $$

Since both \( x \) and \( y \) are negative (\( x = -3 \) and \( y = -4 \)), the vector lies in the third quadrant.

In this quadrant, the vector's direction lies between \( 180^\circ \) and \( 270^\circ \). To calculate it, we add \( \alpha \) to \( 180^\circ \):

$$ \theta = 180^\circ + \alpha = 180^\circ + 53.13^\circ \approx 233.13^\circ $$

The vector's direction is:

$$ \theta \approx 233.13^\circ $$

When plotted on the Cartesian plane, the angle \( 233.13^\circ \) is measured counterclockwise from the positive x-axis.

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Example 4

Finally, consider the case where the vector lies in the fourth quadrant. Here, the components are \( x = 3 \) and \( y = -4 \).

Start by calculating the ratio of the absolute values:

$$ \frac{|y|}{|x|} = \frac{4}{3} \approx 1.333 $$

The acute angle \( \alpha \), relative to the positive x-axis, is:

$$ \alpha = \arctan\left(\frac{|y|}{|x|}\right) = \arctan(1.333) $$

Using a calculator, we find:

$$ \alpha \approx 53.13^\circ $$

Since \( x = 3 \) (positive) and \( y = -4 \) (negative), the vector lies in the fourth quadrant.

In this case, the direction can be measured as a negative angle (below the positive x-axis) or as a positive angle greater than \( 270^\circ \), depending on the convention. For example, subtract \( \alpha \) from \( 360^\circ \):

$$ \theta = 360^\circ - \alpha = 360^\circ - 53.13^\circ \approx 306.87^\circ $$

The vector's direction is:

$$ \theta \approx 306.87^\circ $$

When plotted, the angle \( 306.87^\circ \) is measured counterclockwise from the positive x-axis, indicating the vector's direction.

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Comparison of Cases

  • First quadrant (\( x = 3 \), \( y = 4 \)): Angle \( \approx 53.13^\circ \)
  • Second quadrant (\( x = -3 \), \( y = 4 \)): Angle \( \approx 126.87^\circ \)
  • Third quadrant (\( x = -3 \), \( y = -4 \)): Angle \( \approx 233.13^\circ \)
  • Fourth quadrant (\( x = 3 \), \( y = -4 \)): \( \theta \approx 306.87^\circ \)

These examples clearly demonstrate that a vector's direction depends on both the tangent of its acute angle and the quadrant determined by the signs of \( x \) and \( y \).

So remember, correctly identifying the quadrant is essential for obtaining an angle that adheres to mathematical conventions.




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