Problem 34 - Precalculus with Calculus Previews

Question

Express the given vector (a) in trigonometric form and (b) as a linear combination of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). $$ \langle 7,7 \sqrt{3}\rangle $$

Step-by-Step Solution

Answer

(a) Trigonometric form: \( 14 (\cos\frac{\pi}{3} \mathbf{i} + \sin\frac{\pi}{3} \mathbf{j}) \); (b) Linear combination: \( 7\mathbf{i} + 7\sqrt{3}\mathbf{j} \).

Step 1: Identify Components of the Vector

The given vector is \( \langle 7, 7 \sqrt{3} \rangle \), where 7 is the horizontal (x) component and \( 7 \sqrt{3} \) is the vertical (y) component.

Step 2: Calculate the Magnitude of the Vector

The magnitude of the vector is given by the formula \( \|\mathbf{v}\| = \sqrt{x^2 + y^2} \). Plugging in the components, we have: \( \|\mathbf{v}\| = \sqrt{7^2 + (7 \sqrt{3})^2} = \sqrt{49 + 147} = \sqrt{196} = 14 \).

Step 3: Determine the Angle of the Vector

Find the angle \( \theta \) using the tangent function: \( \tan \theta = \frac{y}{x} = \frac{7 \sqrt{3}}{7} = \sqrt{3} \). The angle \( \theta \) whose tangent is \( \sqrt{3} \) is 60 degrees or \( \frac{\pi}{3} \) radians.

Step 4: Express the Vector in Trigonometric Form

Using the magnitude and angle, the trigonometric form of the vector is \( 14 (\cos\frac{\pi}{3} \mathbf{i} + \sin\frac{\pi}{3} \mathbf{j}) \).

Step 5: Express the Vector as a Linear Combination of \(\mathbf{i}\) and \(\mathbf{j}\)

The vector can be expressed as a linear combination as \( 7\mathbf{i} + 7\sqrt{3}\mathbf{j} \) based on its components.

Key Concepts

Trigonometric Form

Trigonometric form is a way of representing vectors that aligns with circular motion concepts. It uses a vector's magnitude and angle to express its position. This form makes it easy to interpret how a vector acts in systems involving rotation or periodic patterns. To convert a vector to trigonometric form, first determine its magnitude, which gives you an idea of its length. Then calculate the angle of the vector using trigonometric functions like tangent, sine, and cosine. The vector is finally expressed as \[ r (\cos \theta \mathbf{i} + \sin \theta \mathbf{j}) \text{ or } r \operatorname{cis}(\theta) \]where \( r \) is the vector's magnitude, and \( \theta \) is the angle.

Unit Vectors

Unit vectors are fundamental components in vector mathematics. They have a magnitude of 1 and point in a specific direction. The most common unit vectors in a plane are \( \mathbf{i} \) and \( \mathbf{j} \).

• \( \mathbf{i} \) is the unit vector in the direction of the x-axis, represented as \( \langle 1, 0 \rangle \).
• \( \mathbf{j} \) is the unit vector in the direction of the y-axis, represented as \( \langle 0, 1 \rangle \).

Expressing a vector as a combination of these unit vectors makes it easy to understand its behavior in terms of its horizontal (x-axis) and vertical (y-axis) components. For example, the vector \( \langle 7, 7\sqrt{3} \rangle \) is expressed as \( 7 \mathbf{i} + 7\sqrt{3} \mathbf{j} \). This shows how far and in what direction each component of the vector reaches.

Magnitude of a Vector

The magnitude of a vector is its length. It is calculated using the Pythagorean theorem, which combines both components of the vector. Given a vector \( \langle x, y \rangle \), its magnitude is:\[ \|\mathbf{v}\| = \sqrt{x^2 + y^2} \] This formula allows you to measure how strong or long the vector is, regardless of its direction. For instance, for the vector \( \langle 7, 7\sqrt{3} \rangle \), its magnitude is found by:

• Calculate \( x^2 + y^2 \).
• Take the square root of the sum.

This calculation gives a magnitude of 14, meaning the vector stretches 14 units long in its resultant direction.

Angle of a Vector

The angle of a vector is the direction it points concerning the positive x-axis. It shows the orientation of the vector in a plane. Using trigonometry,\[ \tan \theta = \frac{y}{x} \]lets us find \( \theta \). This formula finds the angle

• Use the tangent ratio to relate y and x.
• Find \( \theta \) using inverse tangent functions (\( \tan^{-1} \)).

For vector \( \langle 7, 7\sqrt{3} \rangle \), the angle's tangent is \( \sqrt{3} \) leading to an angle \( \theta \) of 60 degrees or \( \frac{\pi}{3} \) radians. This angle is integral in expressing the vector in trigonometric form, defining its precise direction.