Key Concepts and Formulas

• Median of a Triangle: A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.
• Vector Representation of a Median: If $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are two vectors representing sides of a triangle originating from vertex $A$, the vector representing the median from $A$ to the midpoint of $BC$, denoted as $\overrightarrow{AM}$, is given by $\overrightarrow{AM} = \frac{1}{2}(\overrightarrow{AB} + \overrightarrow{AC})$.
• Magnitude of a Vector: The magnitude (or length) of a vector $\vec{v} = x\widehat{i} + y\widehat{j} + z\widehat{k}$ is given by $|\vec{v}| = \sqrt{x^2 + y^2 + z^2}$.

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Step-by-Step Solution

1. Identify the Given Vectors: We are provided with two vectors that represent the sides of triangle $ABC$ originating from vertex $A$:

   • $\overrightarrow{AB} = 3\widehat{i} + 4\widehat{k}$
   • $\overrightarrow{AC} = 5\widehat{i} - 2\widehat{j} + 4\widehat{k}$ We can explicitly write the components of $\overrightarrow{AB}$ as $3\widehat{i} + 0\widehat{j} + 4\widehat{k}$ for clarity in vector addition.

2. Determine the Vector Representing the Median from A: Let $M$ be the midpoint of the side $BC$. The median through vertex $A$ is the line segment $AM$. In vector form, the vector representing this median is $\overrightarrow{AM}$. Using the formula for the median vector originating from a common vertex: $\overrightarrow{AM} = \frac{1}{2} (\overrightarrow{AB} + \overrightarrow{AC})$ This formula is derived from the midpoint formula in vector form. If $A$ is the origin, then the position vector of $M$ is the average of the position vectors of $B$ and $C$. When $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are given, they correspond to the position vectors of $B$ and $C$ relative to $A$.

3. Substitute the Given Vectors into the Median Formula: We substitute the given vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ into the formula derived in Step 2: $\overrightarrow{AM} = \frac{1}{2} \left( (3\widehat{i} + 0\widehat{j} + 4\widehat{k}) + (5\widehat{i} - 2\widehat{j} + 4\widehat{k}) \right)$

4. Perform Vector Addition: Add the corresponding components of the vectors inside the parentheses: $\overrightarrow{AM} = \frac{1}{2} \left( (3+5)\widehat{i} + (0-2)\widehat{j} + (4+4)\widehat{k} \right)$ $\overrightarrow{AM} = \frac{1}{2} \left( 8\widehat{i} - 2\widehat{j} + 8\widehat{k} \right)$

5. Multiply the Resulting Vector by the Scalar $\frac{1}{2}$: Distribute the scalar $\frac{1}{2}$ to each component of the vector obtained in Step 4: $\overrightarrow{AM} = \left(\frac{8}{2}\right)\widehat{i} + \left(\frac{-2}{2}\right)\widehat{j} + \left(\frac{8}{2}\right)\widehat{k}$ $\overrightarrow{AM} = 4\widehat{i} - 1\widehat{j} + 4\widehat{k}$ This is the vector representing the median from vertex $A$.

6. Calculate the Length (Magnitude) of the Median Vector: The length of the median is the magnitude of the vector $\overrightarrow{AM}$. We use the formula for the magnitude of a vector: $|\overrightarrow{AM}| = \sqrt{(x)^2 + (y)^2 + (z)^2}$ Substituting the components of $\overrightarrow{AM}$: $|\overrightarrow{AM}| = \sqrt{(4)^2 + (-1)^2 + (4)^2}$ $|\overrightarrow{AM}| = \sqrt{16 + 1 + 16}$ $|\overrightarrow{AM}| = \sqrt{33}$

Common Mistakes & Tips

• Incorrect Median Formula: Ensure you are using the correct formula for the median vector, especially when the given vectors originate from the same vertex. Avoid formulas for medians from other vertices.
• Algebraic Errors: Pay close attention to signs and arithmetic during vector addition and scalar multiplication. A small error can lead to an incorrect final answer.
• Magnitude Calculation: Double-check the squaring and summing of components when calculating the magnitude. Remember to include the squares of all components, including the $\widehat{j}$ component, even if it appears to be zero in the initial problem statement.

Summary

The problem asks for the length of the median through vertex $A$ of a triangle $ABC$, given the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$. We first established the vector formula for the median $\overrightarrow{AM}$ originating from $A$ as $\frac{1}{2}(\overrightarrow{AB} + \overrightarrow{AC})$. By substituting the given vectors and performing vector addition and scalar multiplication, we found the median vector to be $4\widehat{i} - \widehat{j} + 4\widehat{k}$. Finally, we calculated the magnitude of this vector, which represents the length of the median, yielding $\sqrt{33}$.

The final answer is \boxed{\sqrt{33}}.