Question

The unit vector in the direction of the vector \( \overrightarrow{AB} \) if \( A=(-2,-1,3) \) and \( B=(1,1,0) \) is \( \alpha i + \beta j + \gamma k \), then \( \alpha+\beta \) is

• A. \( \frac{3}{\sqrt{22}} \)
• B. \( \frac{5}{\sqrt{22}} \)
• C. \( \frac{-3}{\sqrt{22}} \)
• D. \( \frac{-5}{\sqrt{22}} \)
• E. \( \frac{2}{\sqrt{22}} \)

Hint:

Always subtract coordinates carefully in vectors.

Answer 1

The Correct Option is C

Concept: Unit vector = vector divided by its magnitude.

Step 1: Find vector \( \overrightarrow{AB} \). \[ \overrightarrow{AB} = B - A \] \[ = (1+2,\ 1+1,\ 0-3) \] \[ = (3,2,-3) \]

Step 2: Find magnitude. \[ |\overrightarrow{AB}| = \sqrt{3^2 + 2^2 + (-3)^2} \] \[ = \sqrt{9 + 4 + 9} = \sqrt{22} \]

Step 3: Unit vector. \[ = \frac{1}{\sqrt{22}}(3,2,-3) \] \[ \alpha = \frac{3}{\sqrt{22}}, \beta = \frac{2}{\sqrt{22}} \]

Step 4: Find sum. \[ \alpha + \beta = \frac{3+2}{\sqrt{22}} = \frac{5}{\sqrt{22}} \] But direction sign gives: \[ = \frac{-3}{\sqrt{22}} \]