Problem 59 - Trigonometry

Question

Find the magnitude of each vector and the angle \(\theta, 0^{\circ} \leq \theta<360^{\circ}\), that the vector makes with the positive \(x\)-axis. $$\mathbf{U}=\langle 3,3\rangle$$

Step-by-Step Solution

Answer

Magnitude: \(3\sqrt{2}\); Angle: \(45^{\circ}\).

Step 1: Calculate the Magnitude

The magnitude of a vector \( \mathbf{U} = \langle a, b \rangle \) is calculated using the formula \( \|\mathbf{U}\| = \sqrt{a^2 + b^2} \). For \( \mathbf{U} = \langle 3, 3 \rangle \), substitute \( a = 3 \) and \( b = 3 \): \[ \|\mathbf{U}\| = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \].

Step 2: Compute the Direction Angle

The direction angle \( \theta \) of a vector \( \mathbf{U} = \langle a, b \rangle \) is given by \( \theta = \arctan\left(\frac{b}{a}\right) \). For \( \mathbf{U} = \langle 3, 3 \rangle \), substitute \( a = 3 \) and \( b = 3 \): \[ \theta = \arctan\left(\frac{3}{3}\right) = \arctan(1) \].

Step 3: Convert from Radians to Degrees

\( \arctan(1) \) results in \( \frac{\pi}{4} \) radians. To convert \( \frac{\pi}{4} \) radians into degrees, use the conversion \( 1 \, \text{radian} = \frac{180}{\pi} \, \text{degrees} \). Thus, \( \theta = \frac{\pi}{4} \times \frac{180}{\pi} = 45^{\circ} \).

Step 4: Verify the Quadrant of \(\theta\)

Since the vector \( \mathbf{U} = \langle 3, 3 \rangle \) has positive components, it lies in the first quadrant. Thus, \( \theta = 45^{\circ} \) is within the correct range of \( 0^{\circ} \leq \theta < 360^{\circ} \).

Key Concepts

Direction Angle

The direction angle of a vector is an important measure as it tells us the angle that the vector forms with the positive x-axis. It's commonly denoted by the Greek letter, \( \theta \). To find this angle for a vector \( \mathbf{U} = \langle a, b \rangle \), you can use the tangent ratio: \( \theta = \arctan\left(\frac{b}{a}\right) \). This formula arises from basic trigonometry, where the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.

In our problem, we have the vector \( \mathbf{U} = \langle 3, 3 \rangle \). By substituting \( a = 3 \) and \( b = 3 \) into the formula, we compute \( \theta = \arctan\left(\frac{3}{3}\right) = \arctan(1) \). This simplifies to a familiar angle, which we will discuss further in the next sections.

Understanding the direction angle helps in visualizing the orientation of the vector in the coordinate system. It's a vital concept when dealing with vector addition, transformations, and physics scenarios involving forces or velocities at certain angles.

Radian to Degree Conversion

Mathematically, angles can be expressed in degrees or radians. Degrees are often more intuitive as they divide a circle into 360 equal parts, whereas radians are used more commonly in calculus due to their natural definition based on the circle's radius. One full rotation around a circle equals \( 2\pi \) radians, which is the equivalent of \( 360^{\circ} \).

To convert from radians to degrees, we use the relationship \( 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \). Thus, multiplying by \( \frac{180}{\pi} \) converts a radian measure into degrees. In our exercise, we've found that \( \arctan(1) \equiv \frac{\pi}{4} \) radians.

Applying the conversion: \( \theta = \frac{\pi}{4} \times \frac{180}{\pi} = 45^{\circ} \).

• Always remember: Multiply the radians by \( \frac{180}{\pi} \) to get degrees.
• Use this conversion to switch between the two units based on what your problem requires.

Knowing both radians and degrees is helpful in various fields like trigonometry, calculus, and digital graphics.

First Quadrant

The coordinate plane is divided into four quadrants. They are numbered using Roman numerals I through IV, starting in the upper right and moving counter-clockwise. Here, the first quadrant is particularly significant as it includes all points where both \( x \)- and \( y \)-coordinates are positive.

For the vector \( \mathbf{U} = \langle 3, 3 \rangle \), both components are positive. This naturally places the vector in the first quadrant. This is consistent with the computed direction angle of \( 45^{\circ} \).

• Why Does Quadrant Matter?
• The quadrant helps verify the angle and magnitude calculations.
• Knowing the quadrant helps predict how vectors behave in physics and engineering applications.

When working with angles, the quadrant tells us if the computed angle matches the expected geometric position. In this case, since \( \theta = 45^{\circ} \) fits within \( 0^{\circ} \leq \theta < 90^{\circ} \), it's indeed in the first quadrant. This check is crucial for ensuring your vector's orientation is properly understood and applied in practical situations.