If
v
=7
i
–8
j
, what is the magnitude of 4
v
?
Simplify any radicals.

We are given the vector \( \mathbf{v} = 7\mathbf{i} - 8\mathbf{j} \) and are asked to find the magnitude of \( 4\mathbf{v} \).

### Step 1: Express \( 4\mathbf{v} \)
Since \( \mathbf{v} = 7\mathbf{i} - 8\mathbf{j} \), multiplying by 4 gives:
\[
4\mathbf{v} = 4(7\mathbf{i} - 8\mathbf{j}) = 28\mathbf{i} - 32\mathbf{j}.
\]

### Step 2: Find the magnitude of \( 4\mathbf{v} \)
The magnitude of a vector \( a\mathbf{i} + b\mathbf{j} \) is given by:
\[
|\mathbf{v}| = \sqrt{a^2 + b^2}.
\]
In this case, \( a = 28 \) and \( b = -32 \), so the magnitude is:
\[
|4\mathbf{v}| = \sqrt{28^2 + (-32)^2} = \sqrt{784 + 1024} = \sqrt{1808}.
\]

### Step 3: Simplify \( \sqrt{1808} \)
We simplify \( \sqrt{1808} \) by factoring it:
\[
1808 = 16 \times 113.
\]
Thus:
\[
\sqrt{1808} = \sqrt{16 \times 113} = 4\sqrt{113}.
\]

So, the magnitude of \( 4\mathbf{v} \) is:
\[
|4\mathbf{v}| = 4\sqrt{113}.
\]

Would you like more details or have any questions?

Here are some related questions:
1. How do you find the magnitude of a vector in 3D space?
2. What is the significance of unit vectors in vector operations?
3. How can you calculate the direction of a vector from its components?
4. What happens if you multiply a vector by a negative scalar?
5. How can vector magnitudes be applied in physics problems?

**Tip:** Multiplying a vector by a scalar scales both the components and the magnitude of the vector by that scalar.

If v = 7i – 8j, what is the magnitude of 4v? Simplify any radicals.

Explore how to calculate the magnitude of a scaled vector using the example v = 7i – 8j. This step-by-step guide simplifies the process, including scalar multiplication and radical simplification. Ideal for high school students in Grades 9-12 studying vector operations.

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