Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in \(\left[0,360^{\circ}\right)\). $$\langle 1,1\rangle$$

Step-by-Step Solution

Answer

Magnitude: \(\approx 1.4\), Direction: \(45^{\circ}\).

Step 1: Understand the Vector Components

The given vector is \( \langle 1, 1 \rangle \), which means it has an x-component of 1 and a y-component of 1.

Step 2: Calculate the Magnitude of the Vector

The magnitude of a vector \( \langle a, b \rangle \) is calculated using the formula \[ \text{Magnitude} = \sqrt{a^2 + b^2} \].Substitute \( a = 1 \) and \( b = 1 \) into the formula to get:\[\text{Magnitude} = \sqrt{1^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.4.\]

Step 3: Calculate the Direction Angle

The direction angle \( \theta \) of a vector \( \langle a, b \rangle \) can be found using the formula: \[ \theta = \arctan\left(\frac{b}{a}\right) \]. In our case, \( a = 1 \) and \( b = 1 \), so:\[\theta = \arctan\left(\frac{1}{1}\right) = \arctan(1).\]\( \arctan(1) \) results in an angle of \( 45^{\circ} \), since both components are positive and it is in the first quadrant.

Key Concepts

Magnitude

The magnitude of a vector is akin to its length. It's an important measure as it tells you how "long" the vector is without considering its direction. When you're dealing with a vector such as \( \langle 1, 1 \rangle \), the magnitude helps in understanding the scale of influence this vector has in a coordinate system.
To determine the magnitude from vector components \( \langle a, b \rangle \), you use the formula:

• \( \text{Magnitude} = \sqrt{a^2 + b^2} \)

This formula is founded on the Pythagorean theorem, reflecting how vectors represent diagonal positions in a right-angled triangle. By squaring and summing the components \( a \) and \( b \), which serve as the triangle's sides, and then taking the square root, you get the hypotenuse, which reveals the vector's length. In our example, with components \( 1 \) and \( 1 \), you calculate the magnitude as follows:
\[ \text{Magnitude} = \sqrt{1^2 + 1^2} = \sqrt{2} \approx 1.4 \].
This result tells you that the vector, regardless of direction, stretches about 1.4 units in length.

Direction Angle

The direction angle of a vector provides insight into the vector’s orientation within the two-dimensional plane. It specifies where the vector is pointing, measured counterclockwise from the positive x-axis.
For the vector \( \langle 1, 1 \rangle \), calculating the direction angle involves understanding the ratio of its components. The formula:

• \( \theta = \arctan\left(\frac{b}{a}\right) \)

tells us that by applying the inverse tangent (arctan) function to the ratio of the y-component \( b \) over the x-component \( a \), we determine the angle the vector makes with the horizontal axis.
For the vector \( \langle 1, 1 \rangle \), both components are equal and positive, leading us to compute:
\[ \theta = \arctan\left(\frac{1}{1}\right) = \arctan(1) \].
This results in a direction angle of \( 45^{\circ} \). The angle lies within the first quadrant, highlighting that both vector components push equally up and to the right.

Angle Measurement

Angle measurement in vectors is crucial when positioning or transforming vectors. Here, angles are measured in degrees, typically ranging from \( 0^{\circ} \) to \( 360^{\circ} \). Understanding how these angles fit within this framework is key to ensuring directional accuracy of vectors in analysis.
When calculating angles like the one found for vector \( \langle 1, 1 \rangle \), the result is simply \( 45^{\circ} \). Since the vector exists in the first quadrant —where both x and y components are positive— this measurement confirms that the angle aligns well within the \( 0^{\circ} \) to \( 90^{\circ} \) range. Such angles provide essential guidance for identifying precise directions, useful in sectors like navigation, physics, and engineering.

• First Quadrant: \( 0^{\circ} \) to \( 90^{\circ} \)
• Second Quadrant: \( 90^{\circ} \) to \( 180^{\circ} \)
• Third Quadrant: \( 180^{\circ} \) to \( 270^{\circ} \)
• Fourth Quadrant: \( 270^{\circ} \) to \( 360^{\circ} \)

Thus, knowing your angle measurement not only informs you of a vector's direct path but also helps in identifying the quadrant it resides within for practical application.