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

Two vectors are perpendicular if their dot product is zero. If we have two vectors $extbf{a} = a_1 extbf{i} + a_2 extbf{j} + a_3 extbf{k}$ and $extbf{b} = b_1 extbf{i} + b_2 extbf{j} + b_3 extbf{k}$, then the dot product is given by: extbf{a} ullet extbf{b} = a_1b_1 + a_2b_2 + a_3b_3

To find a vector that is perpendicular to $3 extbf{i} + 2 extbf{j} - 5 extbf{k}$, we can check each option.

Calculate the dot product:

& = -3 - 2 - 5 \\ & = -10 \\ ext{(Not perpendicular)} ext{.} \\ \\ \\ \\ \\$

Calculate the dot product:

& = 3 + 2 + 5 \\ & = 10 \\ ext{(Not perpendicular)} ext{.} \\ \\ \\$

Calculate the dot product:

& = -6 + 6 - 5 \\ & = -5 \\ ext{(Not perpendicular)} ext{.} \\$

Calculate the dot product:

& = 9 - 4 - 5 \\ & = 0 \\ ext{(Perpendicular)} ext{.} \\$