Question

If $(2\hat{i} + 3\hat{j} + \hat{k})$ and $(2\hat{i} + \hat{j} - \hat{k})$ represent the sides $\vec{AB}$ and $\vec{AC}$ respectively of $\Delta ABC$, find the vector representing the median through A.

Hint:

This formula is an application of the Parallelogram Law of Vector Addition. The sum $\vec{AB} + \vec{AC}$ gives the diagonal of the completed parallelogram, and the median is exactly half of that diagonal length.

Answer 1

Concept: In a triangle $ABC$, let $D$ be the midpoint of the side $BC$. The line segment $AD$ is the median originating from vertex $A$. By utilizing vector geometry principles, the median vector $\vec{AD}$ can be expressed directly in terms of the two side vectors sharing the same initial point $A$.
• Midpoint Vector Property: If $\vec{AB}$ and $\vec{AC}$ are the two side vectors bounding vertex $A$, then the median vector $\vec{AD}$ is given by the formula: \[ \vec{AD} = \frac{\vec{AB} + \vec{AC}}{2} \]

Step 1: Substitute the given vectors into the median formula and simplify.
We are given: \[ \vec{AB} = 2\hat{i} + 3\hat{j} + \hat{k} \] \[ \vec{AC} = 2\hat{i} + \hat{j} - \hat{k} \] Let $\vec{AD}$ represent the median passing through vertex $A$. Applying the vector addition midpoint theorem: \[ \vec{AD} = \frac{(2\hat{i} + 3\hat{j} + \hat{k}) + (2\hat{i} + \hat{j} - \hat{k})}{2} \] Combine the like scalar components ($\hat{i}$, $\hat{j}$, and $\hat{k}$ terms) in the numerator: \[ \vec{AD} = \frac{(2+2)\hat{i} + (3+1)\hat{j} + (1-1)\hat{k}}{2} \] \[ \vec{AD} = \frac{4\hat{i} + 4\hat{j} + 0\hat{k}}{2} \]

Step 2: Perform scalar division to obtain the final simplified components.
Divide each component by 2: \[ \vec{AD} = \frac{4}{2}\hat{i} + \frac{4}{2}\hat{j} = 2\hat{i} + 2\hat{j} \] The vector representing the median through $A$ is $2\hat{i} + 2\hat{j}$.