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Question
The vectors \[2 \hat{i} + 3 \hat{j} - 4 \hat{k}\] and \[a \hat{i} + \hat{b} j + c \hat{k}\] are perpendicular if
Options
(a) a = 2, b = 3, c = −4
(b) a = 4, b = 4, c = 5
(c) a = 4, b = 4, c = −5
(d) a = −4, b = 4, c = −5
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Solution
(b) a = 4, b = 4, c = 5
\[\text{ It is given that vectors } 2 \hat{i} + 3 \hat{j} - 4 \hat{k} \text{ and } a \text{i} + b \hat{j} + c \hat{k} \text{ are perpendicular }.\]
\[\text{ So, their dot product is zero }.\]
\[ \Rightarrow 2a + 3b - 4c = 0\]
\[\left( b \right) a = 4; b = 4; c = 5\]
\[ \Rightarrow 2\left( 4 \right) + 3\left( 4 \right) - 4\left( 5 \right) = 0\]
\[8 + 12 - 20 = 0\]
\[0 = 0, \text{ which is true }.\]
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