The vector c directed along the bisectors of the angle...

The vector c directed along the bisectors of the angle between the vectors $a=7 i-4 j-4 k$ and $\hat{b}=-2 i-j+2 k$ if $|c|=3 \sqrt{6}$, is given by
(A) $i-7 j+2 k$
(B) $2 i+7 j-3 k$
(C) $-i+7 j-2 k$
(D) $4 i+7 j-4 k$

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Key Concepts

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar, which scales the magnitude of the vector while preserving its direction. This is particularly useful when a vector calculated in a certain direction (such as an angle bisector) needs to be adjusted to have a specified length.

Vector Magnitude

The magnitude (or length) of a vector is computed as the square root of the sum of the squares of its components. This concept is essential when normalizing vectors, ensuring that the unit vectors used in many vector operations, including the construction of an angle bisector, are correctly scaled.

Vector Angle Bisector

The angle bisector of two vectors is a vector that divides the angle between the two given vectors into two equal angles. In vector analysis, when the bisector is required to have a specific magnitude, the process involves finding the unit vectors along the given vectors and then summing these unit vectors to obtain a direction that bisects the angle.

Unit Vector

A unit vector is a vector that has a magnitude of one and is used to indicate direction only. By dividing any vector by its magnitude, one obtains its corresponding unit vector, which is crucial for standardizing vectors before performing operations like finding an angle bisector.

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