Determine the magnitude, and direction in degrees of
u = <minus−3,7> in the interval
0 less than or equal to≤ thetaθ < 360degrees°.
Round to three decimal places when necessary.

To determine the magnitude and direction of the vector \( \mathbf{u} = \langle -3, 7 \rangle \), we will follow these steps:

### Step 1: Calculate the Magnitude
The magnitude \( |\mathbf{u}| \) of a vector \( \mathbf{u} = \langle x, y \rangle \) is given by:

\[
|\mathbf{u}| = \sqrt{x^2 + y^2}
\]

For the vector \( \mathbf{u} = \langle -3, 7 \rangle \):

\[
|\mathbf{u}| = \sqrt{(-3)^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58} \approx 7.616
\]

So, the magnitude is approximately \( 7.616 \).

### Step 2: Calculate the Direction (Angle)
The direction \( \theta \) of the vector is the angle it makes with the positive \( x \)-axis. This can be found using the formula for the tangent of the angle:

\[
\tan \theta = \frac{y}{x}
\]

For \( \mathbf{u} = \langle -3, 7 \rangle \):

\[
\tan \theta = \frac{7}{-3}
\]

Now, we compute the inverse tangent (arctangent) of this ratio:

\[
\theta = \tan^{-1} \left( \frac{7}{-3} \right)
\]

Using a calculator, \( \tan^{-1} \left( \frac{7}{-3} \right) \approx -66.802^\circ \).

Since the vector lies in the second quadrant (because the \( x \)-component is negative and the \( y \)-component is positive), we need to adjust this angle by adding \( 180^\circ \) to ensure it's within the interval \( [0^\circ, 360^\circ) \).

\[
\theta = 180^\circ - 66.802^\circ = 113.198^\circ
\]

Thus, the direction of the vector is approximately \( 113.198^\circ \).

### Final Answer:
- **Magnitude**: \( 7.616 \)
- **Direction**: \( 113.198^\circ \)

Would you like more details on any part of the solution, or have further questions?

### 5 Related Questions:
1. How would the magnitude change if the vector components were doubled?
2. What is the direction of the vector \( \langle -7, 3 \rangle \)?
3. How do you convert polar coordinates to rectangular coordinates?
4. How would you compute the magnitude and direction for a 3D vector?
5. How do you find the angle between two vectors?

### Tip:
Always remember to check the quadrant of the vector when determining the angle, as arctangent gives values that may need adjustment based on the signs of the components.

Determine the magnitude, and direction in degrees of u = <−3,7> in the interval 0 ≤ θ < 360°. Round to three decimal places when necessary.

Learn how to calculate the magnitude and direction of vector u = <-3,7>. The solution includes step-by-step instructions on using the Pythagorean theorem to find the magnitude and the inverse tangent function to determine the angle. Suitable for high school students.

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