Which of the following vectors is perpendicular to 3 extbf{i} + 2 extbf{j} - 5 extbf{k}? A - HSC - SSCE Mathematics Extension 2 - Question 1 - 2024 - Paper 1

To find a vector that is perpendicular to another vector, we can use the dot product. Two vectors ( \textbf{a} ) and ( \textbf{b} ) are perpendicular if ( \textbf{a} , \cdot , \textbf{b} = 0 ).

Now we will evaluate each option provided:

• Option A: ( -\textbf{i} - \textbf{j} + \textbf{k} )

  • Dot product: ( (3)(-1) + (2)(-1) + (-5)(1) = -3 - 2 - 5 = -10 ) (Not perpendicular)

• Option B: ( \textbf{i} + \textbf{j} - \textbf{k} )

  • Dot product: ( (3)(1) + (2)(1) + (-5)(-1) = 3 + 2 + 5 = 10 ) (Not perpendicular)

• Option C: ( -2\textbf{i} + 3\textbf{j} + \textbf{k} )

  • Dot product: ( (3)(-2) + (2)(3) + (-5)(1) = -6 + 6 - 5 = -5 ) (Not perpendicular)

• Option D: ( 3\textbf{i} - 2\textbf{j} + \textbf{k} )

  • Dot product: ( (3)(3) + (2)(-2) + (-5)(1) = 9 - 4 - 5 = 0 ) (Perpendicular)

The vector that is perpendicular to ( 3\textbf{i} + 2\textbf{j} - 5\textbf{k} ) is Option D: ( 3\textbf{i} - 2\textbf{j} + \textbf{k} ).