Vectors and are perpendicular to each other when 3a + 2b = 7, the ratio of a to b is . The value of x is ________ . [2023]

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Solution

Q.
Vectors $a \hat{i} + b \hat{j} + \hat{k}$ and $2 \hat{i} - 3 \hat{j} + 4 \hat{k}$ are perpendicular to each other when 3a + 2b = 7, the ratio of a to b is $\frac{x}{2}$ . The value of x is ________ . [2023]

Ans.
(1)

For two perpendicular vectors

$(a \hat{i} + b \hat{j} + \hat{k}) \cdot (2 \hat{i} - 3 \hat{j} + 4 \hat{k}) = 0$

2a – 3b + 4 = 0

on solving, 2a – 3b = –4

also given 3a + 2b = 7

We get a = 1, b = 2

$\frac{a}{b} = \frac{x}{2} \Rightarrow x = \frac{2 a}{b} = \frac{2 \times 1}{2} = 1$

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