https://file.aiquickdraw.com/mathbot/file/1730419308381yjpge6t2.jpg

To find a vector \(\vec{v}\) that is parallel to \(\vec{u} = -\hat{i} + 6\hat{j} - 3\hat{k}\) and has a magnitude of \(|\vec{v}| = 4\), we can follow these steps:

1. **Calculate the magnitude of \(\vec{u}\):**
   \[
   |\vec{u}| = \sqrt{(-1)^2 + 6^2 + (-3)^2} = \sqrt{1 + 36 + 9} = \sqrt{46}.
   \]

2. **Find the unit vector in the direction of \(\vec{u}\):**
   The unit vector \(\hat{u}\) in the direction of \(\vec{u}\) is given by:
   \[
   \hat{u} = \frac{\vec{u}}{|\vec{u}|} = \frac{-\hat{i} + 6\hat{j} - 3\hat{k}}{\sqrt{46}}.
   \]

3. **Scale the unit vector to have a magnitude of 4:**
   Since \(\vec{v}\) is parallel to \(\vec{u}\), we can write \(\vec{v} = k \hat{u}\) where \(k\) is chosen so that \(|\vec{v}| = 4\). Thus,
   \[
   \vec{v} = 4 \cdot \frac{-\hat{i} + 6\hat{j} - 3\hat{k}}{\sqrt{46}}.
   \]

4. **Simplify the expression:**
   Distribute the \(4\) in the numerator:
   \[
   \vec{v} = \frac{-4\hat{i} + 24\hat{j} - 12\hat{k}}{\sqrt{46}}.
   \]

So, the answer is:
\[
\vec{v} = \frac{-4\hat{i} + 24\hat{j} - 12\hat{k}}{\sqrt{46}}.
\]

Would you like more details on any of these steps, or do you have further questions?

---

1. How do we calculate the magnitude of a vector?
2. What is a unit vector and why is it useful?
3. Why is \(\vec{v}\) parallel to \(\vec{u}\) if it is a scalar multiple of \(\vec{u}\)?
4. How do we determine the scalar factor to adjust vector magnitude?
5. Can this method be applied to any vector with a specified magnitude?

**Tip:** A vector is parallel to another if it’s a scalar multiple of it, which keeps their directions aligned.

Find a vector \(\vec{v}\) parallel to \(\vec{u} = -\hat{i} + 6\hat{j} - 3\hat{k}\) and with magnitude \(|\vec{v}| = 4\).

Learn how to find a vector parallel to \(\vec{u} = -\hat{i} + 6\hat{j} - 3\hat{k}\) with a specified magnitude of 4. This solution includes key concepts like unit vectors, magnitude calculation, and scalar multiplication, suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation