Vectors
Vectors
Allen Star Batch
Grade 12
Question:
The vector, directed along the internal bisector of the angle between the vectors $\vec{a} = 7\vec{i} - 4\vec{j} - 4\vec{k}$ & $\vec{b} = -2\vec{i} - \vec{j} + 2\vec{k}$ with $|\vec{c}| = 5\sqrt{6}$ is:
$\frac{5}{3}(\vec{i} - 7\vec{j} + 2\vec{k})$
$\frac{5}{3}(\vec{i} + 7\vec{j} - 2\vec{k})$
$\frac{5}{3}(-\vec{i} - 7\vec{j} + 2\vec{k})$
None of these
Step-by-Step Solution
Key Concept: The internal bisector of angle between vectors $\vec{a}$ and $\vec{b}$ is parallel to $\frac{\vec{a}}{|\vec{a}|} + \frac{\vec{b}}{|\vec{b}|}$. After finding unit vectors and their sum, scale to magnitude $5\sqrt{6}$ using the constraint $|\vec{c}| = 5\sqrt{6}$.
The internal bisector of an angle between two vectors $\vec{a}$ and $\vec{b}$ is parallel to the vector $\frac{\vec{a}}{|\vec{a}|} + \frac{\vec{b}}{|\vec{b}|}$, which simplifies to $\vec{a} + \vec{b}$ when both vectors are unit vectors or when magnitudes are equal.
Correct Answer: 1