vector u has an initial point at (-3, 1) and a terminal point at (-10,7). What is u in trigonometric form?

To express vector \( \mathbf{u} \) in trigonometric form, we need to find its magnitude and direction angle.

### Step 1: Find the components of vector \( \mathbf{u} \)
The vector \( \mathbf{u} \) has an initial point \( P_1(-3, 1) \) and a terminal point \( P_2(-10, 7) \). The components of the vector are found by subtracting the coordinates of the initial point from the terminal point:

\[
\mathbf{u} = (x_2 - x_1, y_2 - y_1) = (-10 - (-3), 7 - 1) = (-7, 6)
\]

So, \( \mathbf{u} = -7\mathbf{i} + 6\mathbf{j} \).

### Step 2: Find the magnitude of \( \mathbf{u} \)
The magnitude \( \|\mathbf{u}\| \) is given by:

\[
\|\mathbf{u}\| = \sqrt{(-7)^2 + 6^2} = \sqrt{49 + 36} = \sqrt{85}
\]

So, the magnitude \( \|\mathbf{u}\| \) is \( \sqrt{85} \).

### Step 3: Find the direction angle \( \theta \)
The direction angle \( \theta \) of the vector with respect to the positive x-axis is given by:

\[
\theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{6}{-7}\right)
\]

Since the vector is in the second quadrant (where the x-component is negative and the y-component is positive), the angle \( \theta \) should be adjusted accordingly. We find the principal value and then adjust for the correct quadrant:

\[
\theta = \tan^{-1}\left(-\frac{6}{7}\right)
\]

This gives an angle in the fourth quadrant, so we add \( 180^\circ \) to place it in the second quadrant:

\[
\theta = 180^\circ + \tan^{-1}\left(-\frac{6}{7}\right)
\]

### Final Answer:
The vector \( \mathbf{u} \) in trigonometric form is:

\[
\mathbf{u} = \sqrt{85} \left(\cos \theta \mathbf{i} + \sin \theta \mathbf{j}\right)
\]

where \( \theta \) is the angle \( 180^\circ + \tan^{-1}\left(-\frac{6}{7}\right) \).

Do you need more details on any of these steps or have any other questions?

Here are 5 related questions to consider:
1. How do you convert a vector from trigonometric form back to component form?
2. What are the different ways to express a vector in polar coordinates?
3. How do you find the unit vector in the direction of \( \mathbf{u} \)?
4. How does the direction angle relate to the components of a vector?
5. What is the significance of the magnitude and direction in physical applications of vectors?

**Tip:** When working with angles in trigonometric functions, always pay attention to the quadrant in which the vector lies to correctly determine the direction angle.

Learn how to express vector u in trigonometric form from its initial point (-3, 1) to its terminal point (-10, 7) using vector components, magnitude, and direction angle formulas. Suitable for high school students studying vectors and trigonometry.

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