Problem 16 - University Calculus: Early Transcendentals

Question

Find the component form of the vector. The unit vector obtained by rotating the vector \langle 1,0\rangle \(135^{\circ}\) counterclockwise about the origin.

Step-by-Step Solution

Answer

The unit vector is \(\langle -\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \rangle\).

Step 1: Understand the Original Vector

The original vector is given as \(\langle 1, 0 \rangle\). This vector points along the positive x-axis and has a magnitude of 1, making it already a unit vector.

Step 2: Rotation Angle Conversion

The problem states a rotation angle of \(135^{\circ}\) counterclockwise. Convert this angle to radians since trigonometric functions typically use radians:\[135^{\circ} = \frac{135 \pi}{180} = \frac{3\pi}{4} \text{ radians}.\]

Step 3: Apply the Rotation Formula

To rotate a vector \(\langle x, y \rangle\) by an angle \(\theta\), use the rotation formulas:\[x' = x \cos(\theta) - y \sin(\theta)\]\[y' = x \sin(\theta) + y \cos(\theta).\]With \(x = 1\), \(y = 0\), and \(\theta = \frac{3\pi}{4}\), substitute these values into the formulas:\[x' = 1 \cdot \cos\left(\frac{3\pi}{4}\right) - 0 \cdot \sin\left(\frac{3\pi}{4}\right) = \cos\left(\frac{3\pi}{4}\right),\]\[y' = 1 \cdot \sin\left(\frac{3\pi}{4}\right) + 0 \cdot \cos\left(\frac{3\pi}{4}\right) = \sin\left(\frac{3\pi}{4}\right).\]

Step 4: Compute Trigonometric Values

Calculate \(\cos\left(\frac{3\pi}{4}\right)\) and \(\sin\left(\frac{3\pi}{4}\right)\):\[\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2},\quad \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}.\]

Step 5: Determine the New Vector Components

Substitute the trigonometric values back into the formulas from Step 3:\[x' = -\frac{\sqrt{2}}{2},\quad y' = \frac{\sqrt{2}}{2}.\]So, the component form of the rotated vector is \(\langle -\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \rangle\).

Key Concepts

Trigonometric Functions

Trigonometric functions are essential tools in vector rotation. These functions include sine (\( \sin \theta \)) and cosine (\( \cos \theta \)), which relate angles to the ratios of sides in a right triangle. When rotating vectors, these functions help us determine the new components of the vector after the rotation.

• The cosine of an angle gives us the ratio of the adjacent side to the hypotenuse.
• The sine gives us the ratio of the opposite side to the hypotenuse.

By applying these functions to vectors, we can calculate the new positions of vector components after rotating by a given angle. This helps us derive the vector's new direction without altering its magnitude for unit vectors.

Radians Conversion

Radians are a crucial unit for measuring angles, necessary for calculations involving trigonometric functions. Most trigonometric functions in mathematics operate under the assumption that angles are given in radians rather than degrees.
To convert degrees to radians, use the formula:
\[\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}\]
For example, an angle of 135° can be converted to radians:\[135^{\circ} = \frac{135 \pi}{180} = \frac{3\pi}{4}\text{ radians}\]This conversion is essential for accurately applying trigonometric functions to rotate vectors.

Unit Vector

A unit vector is a vector with a magnitude of one, serving as a useful representation for direction. When dealing with rotations, unit vectors allow us to focus solely on the orientation aspect while keeping magnitude constant.

• Unit vectors help normalize any vector by dividing each component by the vector's magnitude.
• They are typically represented using angle brackets, like \(\langle 1, 0\rangle\), indicating direction along axes.

The original vector in this exercise is already a unit vector, indicating it points entirely along the x-axis with no y-component before rotation. This makes computing rotations straightforward since we are only concerned with changes in direction.

Vector Components

Vectors are made up of components, which are projections on the coordinate axes. These components define the vector's position and orientation in space.

• The x-component is how far the vector extends along the x-axis.
• The y-component is how far it extends along the y-axis.

To find the new components of a rotated vector, we apply the following rotation transformations:\[x' = x \cos(\theta) - y \sin(\theta)\]\[y' = x \sin(\theta) + y \cos(\theta)\]In this exercise, the original vector \(\langle 1, 0\rangle\) after a 135° rotation becomes:\[ x' = -\frac{\sqrt{2}}{2}, \quad y' = \frac{\sqrt{2}}{2}\]Effectively, this transformation gives us the new vector components reflecting the rotation.