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.