Key Concepts and Formulas

• Vector Representation of a Median: If $D$ is the midpoint of side $BC$ in triangle $ABC$, the vector representing the median from vertex $A$ to $D$ is given by $\overrightarrow{AD} = \frac{\overrightarrow{AB} + \overrightarrow{AC}}{2}$. This formula arises from the midpoint formula for vectors and the triangle law of vector addition.
• Magnitude of a Vector: The length or magnitude of a vector $\vec{v} = x\widehat{i} + y\widehat{j} + z\widehat{k}$ is calculated as $|\vec{v}| = \sqrt{x^2 + y^2 + z^2}$.

Step-by-Step Solution

Step 1: Identify the given side vectors. We are given the vectors representing two sides of triangle $ABC$ originating from vertex $A$: $\overrightarrow{AB} = 3\widehat{i} + 4\widehat{k}$ $\overrightarrow{AC} = 5\widehat{i} - 2\widehat{j} + 4\widehat{k}$ To facilitate calculations, we can explicitly write the $\widehat{j}$ component of $\overrightarrow{AB}$ as 0: $\overrightarrow{AB} = 3\widehat{i} + 0\widehat{j} + 4\widehat{k}$

Step 2: Calculate the vector representing the median through A. The median through vertex $A$ connects $A$ to the midpoint $D$ of the side $BC$. We use the formula for the median vector: $\overrightarrow{AD} = \frac{\overrightarrow{AB} + \overrightarrow{AC}}{2}$ Substitute the given vectors into the formula: $\overrightarrow{AD} = \frac{(3\widehat{i} + 0\widehat{j} + 4\widehat{k}) + (5\widehat{i} - 2\widehat{j} + 4\widehat{k})}{2}$ Add the corresponding components of the vectors: $\overrightarrow{AD} = \frac{(3+5)\widehat{i} + (0-2)\widehat{j} + (4+4)\widehat{k}}{2}$ $\overrightarrow{AD} = \frac{8\widehat{i} - 2\widehat{j} + 8\widehat{k}}{2}$ Divide each component by 2 to obtain the median vector: $\overrightarrow{AD} = 4\widehat{i} - \widehat{j} + 4\widehat{k}$

Step 3: Calculate the length of the median. The length of the median is the magnitude of the vector $\overrightarrow{AD}$. Using the formula for the magnitude of a vector: $|\overrightarrow{AD}| = \sqrt{(4)^2 + (-1)^2 + (4)^2}$ $|\overrightarrow{AD}| = \sqrt{16 + 1 + 16}$ $|\overrightarrow{AD}| = \sqrt{33}$

Common Mistakes & Tips

• Missing Components: Be careful to include all components ($\widehat{i}$, $\widehat{j}$, $\widehat{k}$) in your vector addition, even if a component is zero. Forgetting the $\widehat{j}$ component in $\overrightarrow{AB}$ can lead to an incorrect sum.
• Sign Errors: Pay close attention to the signs of the components when adding or subtracting vectors, especially when dealing with negative coefficients like $-2\widehat{j}$.
• Squaring Errors: When calculating the magnitude, ensure that each component is squared correctly, and that negative signs are handled properly (e.g., $(-1)^2 = 1$).

Summary

To find the length of the median through vertex $A$ of a triangle $ABC$, we first determine the median vector $\overrightarrow{AD}$ using the formula $\overrightarrow{AD} = \frac{\overrightarrow{AB} + \overrightarrow{AC}}{2}$. Once the median vector is found by adding the components of $\overrightarrow{AB}$ and $\overrightarrow{AC}$ and dividing by two, its length is calculated by finding its magnitude using the formula $|\overrightarrow{AD}| = \sqrt{x^2 + y^2 + z^2}$. In this case, the median vector is $4\widehat{i} - \widehat{j} + 4\widehat{k}$, and its length is $\sqrt{33}$.

The final answer is $\boxed{\sqrt{33}}$ which corresponds to option (D).