Convert w=<-84, -13> to trigonometric form

To convert the vector \( \mathbf{w} = \langle -84, -13 \rangle \) to its trigonometric form, we'll follow these steps:

### Step 1: Find the Magnitude
The magnitude \( r \) of the vector \( \mathbf{w} = \langle -84, -13 \rangle \) is given by the formula:
\[
r = \sqrt{x^2 + y^2}
\]
Substituting the values:
\[
r = \sqrt{(-84)^2 + (-13)^2} = \sqrt{7056 + 169} = \sqrt{7225} = 85
\]

### Step 2: Find the Direction Angle
The direction angle \( \theta \) of the vector \( \mathbf{w} \) with respect to the positive x-axis is given by:
\[
\theta = \tan^{-1}\left(\frac{y}{x}\right)
\]
Substituting the values:
\[
\theta = \tan^{-1}\left(\frac{-13}{-84}\right)
\]
Since both components are negative, the angle lies in the third quadrant. First, we calculate the angle in the first quadrant:
\[
\theta_0 = \tan^{-1}\left(\frac{13}{84}\right) \approx \tan^{-1}(0.15476) \approx 8.82^\circ
\]
Now, since it's in the third quadrant:
\[
\theta = 180^\circ + 8.82^\circ = 188.82^\circ
\]

### Trigonometric Form
The trigonometric form of the vector \( \mathbf{w} \) is given by:
\[
\mathbf{w} = r (\cos \theta \hat{i} + \sin \theta \hat{j})
\]
Substituting the values:
\[
\mathbf{w} = 85 \left( \cos 188.82^\circ \hat{i} + \sin 188.82^\circ \hat{j} \right)
\]

So, the vector \( \mathbf{w} = \langle -84, -13 \rangle \) in trigonometric form is:
\[
\mathbf{w} = 85 \left( \cos 188.82^\circ \hat{i} + \sin 188.82^\circ \hat{j} \right)
\]

Would you like more details or have any questions?

Here are five related questions:
1. How would you convert a vector from trigonometric to rectangular form?
2. What is the significance of the magnitude and direction in vector analysis?
3. How do you determine the quadrant of a vector based on its components?
4. Can you convert the angle from degrees to radians for the trigonometric form?
5. What is the effect of multiplying a vector by a scalar on its trigonometric form?

**Tip:** Always ensure that your angle corresponds to the correct quadrant when converting a vector to trigonometric form.

Learn how to convert the vector w = <-84, -13> to its trigonometric form using the magnitude and direction angle. This problem involves key concepts in vectors and trigonometry, including calculating magnitude and determining direction angles. Suitable for high school students studying vectors and trigonometry.

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