u=6i-15j, v=-22i+12j. write u and v in trigonometric form

To write the vectors \( \mathbf{u} = 6\mathbf{i} - 15\mathbf{j} \) and \( \mathbf{v} = -22\mathbf{i} + 12\mathbf{j} \) in trigonometric (or polar) form, we follow these steps:

### Step 1: Find the magnitude of each vector.

For a vector \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \), its magnitude \( |\mathbf{u}| \) is given by:
\[
|\mathbf{u}| = \sqrt{a^2 + b^2}
\]
Similarly, for \( \mathbf{v} = c\mathbf{i} + d\mathbf{j} \), its magnitude \( |\mathbf{v}| \) is:
\[
|\mathbf{v}| = \sqrt{c^2 + d^2}
\]

#### Magnitude of \( \mathbf{u} \):
\[
|\mathbf{u}| = \sqrt{6^2 + (-15)^2} = \sqrt{36 + 225} = \sqrt{261} \approx 16.16
\]

#### Magnitude of \( \mathbf{v} \):
\[
|\mathbf{v}| = \sqrt{(-22)^2 + 12^2} = \sqrt{484 + 144} = \sqrt{628} \approx 25.08
\]

### Step 2: Find the angle of each vector.

The angle \( \theta \) of a vector \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \) with respect to the positive x-axis (the angle it makes in standard position) is given by:
\[
\theta_u = \text{tan}^{-1}\left(\frac{b}{a}\right)
\]
Similarly for \( \mathbf{v} \):
\[
\theta_v = \text{tan}^{-1}\left(\frac{d}{c}\right)
\]

#### Angle of \( \mathbf{u} \):
\[
\theta_u = \text{tan}^{-1}\left(\frac{-15}{6}\right) = \text{tan}^{-1}\left(-2.5\right)
\]
This gives \( \theta_u \approx -68.20^\circ \).

#### Angle of \( \mathbf{v} \):
\[
\theta_v = \text{tan}^{-1}\left(\frac{12}{-22}\right) = \text{tan}^{-1}\left(-0.545\right)
\]
This gives \( \theta_v \approx -28.81^\circ \).

### Step 3: Write in Trigonometric Form

The trigonometric form of a vector is \( |\mathbf{u}| (\cos \theta_u \mathbf{i} + \sin \theta_u \mathbf{j}) \).

#### Trigonometric form of \( \mathbf{u} \):
\[
\mathbf{u} \approx 16.16 (\cos(-68.20^\circ) \mathbf{i} + \sin(-68.20^\circ) \mathbf{j})
\]

#### Trigonometric form of \( \mathbf{v} \):
\[
\mathbf{v} \approx 25.08 (\cos(-28.81^\circ) \mathbf{i} + \sin(-28.81^\circ) \mathbf{j})
\]

Thus, these are the trigonometric forms of the vectors \( \mathbf{u} \) and \( \mathbf{v} \).

Would you like to go deeper into any part of the process or check other examples?

### Related Questions:
1. How do you convert a vector from polar form back to rectangular form?
2. What is the significance of the angle in trigonometric form?
3. How can you add two vectors in polar form?
4. What is the relationship between the magnitude and the direction of a vector?
5. Can trigonometric form be used to calculate dot products or cross products?

### Tip:
When dealing with vectors, always double-check the quadrant of the angle to ensure that the angle corresponds to the correct direction of the vector.

Learn how to convert vectors u = 6i - 15j and v = -22i + 12j into trigonometric form. This problem involves finding the magnitude and angle of each vector and expressing them using trigonometric functions. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation