To find the magnitude of the vector [tex]\(\frac{1}{3} v + 4 v\)[/tex], given that [tex]\(\| v \| = 6\)[/tex]:
### Step 1: Identify Scalar Multiplications
First, note the scalar multipliers involved in the given expression:
- The scalar multiplier for [tex]\(v\)[/tex] in [tex]\(\frac{1}{3}v\)[/tex] is [tex]\(\frac{1}{3}\)[/tex].
- The scalar multiplier for [tex]\(v\)[/tex] in [tex]\(4v\)[/tex] is [tex]\(4\)[/tex].
### Step 2: Combine the Scaled Vectors
Next, we combine the scalar multipliers with the vector [tex]\(v\)[/tex]:
[tex]\[
\frac{1}{3} v + 4 v = \left( \frac{1}{3} + 4 \right) v
\][/tex]
### Step 3: Perform the Addition of Scalars
Combine the scalar values:
[tex]\[
\frac{1}{3} + 4 = \frac{1}{3} + \frac{12}{3} = \frac{13}{3}
\][/tex]
So, we can rewrite the expression as:
[tex]\[
\frac{13}{3} v
\][/tex]
### Step 4: Calculate the Magnitude
The magnitude of a scalar multiple of a vector ([tex]\(k \cdot v\)[/tex]) is [tex]\(|k| \cdot \|v\| \)[/tex]. Therefore, we need to calculate the product of the absolute value of [tex]\(\frac{13}{3}\)[/tex] and the magnitude of [tex]\(v\)[/tex]:
[tex]\[
\left| \frac{13}{3} \right| \cdot \| v \| = \frac{13}{3} \cdot 6
\][/tex]
### Step 5: Simplify the Final Expression
Multiply the two values:
[tex]\[
\frac{13}{3} \times 6 = 13 \times 2 = 26
\][/tex]
Therefore, the magnitude of [tex]\(\frac{1}{3}v + 4v\)[/tex] is:
[tex]\[
\boxed{26}
\][/tex]
