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Solve this question about unit vector topic

To solve this question, let's break down the information provided:

### Step 1: Analyze the Information Given

The question asks to find the **magnitude** of the vector \(5 \vec{a}_x + \vec{N} - 3 \vec{M}\).

We also see:
- Vector \(\vec{M}\) is defined as \( \vec{M} = -10 \vec{a}_x + 4 \vec{a}_y \).
- Vector \(\vec{N}\) is given as \(\vec{N} = 8 \vec{a}_x + 7 \vec{a}_y - 2 \vec{a}_z\).

The vector expression provided is:
\[
5 \vec{a}_x + \vec{N} - 3 \vec{M}
\]

### Step 2: Substitute \(\vec{M}\) and \(\vec{N}\) into the Expression

Using the definitions of \(\vec{M}\) and \(\vec{N}\):

1. Substitute \(\vec{N}\):
   \[
   \vec{N} = 8 \vec{a}_x + 7 \vec{a}_y - 2 \vec{a}_z
   \]

2. Substitute \(\vec{M}\):
   \[
   \vec{M} = -10 \vec{a}_x + 4 \vec{a}_y
   \]

Now substitute these into \(5 \vec{a}_x + \vec{N} - 3 \vec{M}\):

\[
5 \vec{a}_x + (8 \vec{a}_x + 7 \vec{a}_y - 2 \vec{a}_z) - 3 (-10 \vec{a}_x + 4 \vec{a}_y)
\]

### Step 3: Simplify Each Component

Expanding each part:

1. The \(5 \vec{a}_x\) term remains as \(5 \vec{a}_x\).

2. Adding \(\vec{N}\):
   \[
   5 \vec{a}_x + 8 \vec{a}_x + 7 \vec{a}_y - 2 \vec{a}_z = 13 \vec{a}_x + 7 \vec{a}_y - 2 \vec{a}_z
   \]

3. Distributing the \(-3\) across \(\vec{M}\):
   \[
   -3 (-10 \vec{a}_x + 4 \vec{a}_y) = 30 \vec{a}_x - 12 \vec{a}_y
   \]

4. Now add everything together:
   \[
   (13 \vec{a}_x + 7 \vec{a}_y - 2 \vec{a}_z) + (30 \vec{a}_x - 12 \vec{a}_y) = 43 \vec{a}_x - 5 \vec{a}_y - 2 \vec{a}_z
   \]

So the resulting vector is:
\[
43 \vec{a}_x - 5 \vec{a}_y - 2 \vec{a}_z
\]

### Step 4: Find the Magnitude of the Vector

To find the magnitude, we use the formula for the magnitude of a vector \( \vec{V} = V_x \vec{a}_x + V_y \vec{a}_y + V_z \vec{a}_z \):
\[
|\vec{V}| = \sqrt{V_x^2 + V_y^2 + V_z^2}
\]

Here, \( V_x = 43 \), \( V_y = -5 \), and \( V_z = -2 \).

So,
\[
|\vec{V}| = \sqrt{43^2 + (-5)^2 + (-2)^2}
\]
\[
= \sqrt{1849 + 25 + 4}
\]
\[
= \sqrt{1878}
\]
\[
\approx 43.33
\]

### Final Answer

The magnitude of the vector \(5 \vec{a}_x + \vec{N} - 3 \vec{M}\) is approximately:
\[
43.33
\]

Would you like a more detailed explanation on any specific part?

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Here are 5 related questions for further exploration:

1. How would the result change if the direction of one component in \(\vec{M}\) or \(\vec{N}\) was reversed?
2. What is the importance of unit vectors in vector operations?
3. How can we determine if two vectors are perpendicular or parallel using their components?
4. How does vector addition differ from scalar addition?
5. What are some real-world applications where vector magnitude is essential?

**Tip:** Always break down vector operations step-by-step, especially when multiple vectors are involved, to avoid confusion and errors.

Learn how to find the magnitude of a vector expression involving unit vectors. This problem explores vector addition, scalar multiplication, and magnitude calculations. Suitable for students in Grades 11-12.

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