Vectors a i ^ + b j ^ + k ^ and 2 i ^ - 3 j ^ + 4 k ^ are perpendicular to each other when 3 a + 2 b = 7 , the ratio of a to b is x 2 . The value of x is _____.

Question

Aicharya · Accepted Answer

Dot product of two perpendicular vectors will be zero. Therefore,

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

$\Rightarrow 2 a - 3 b + 4 = 0 . . . E q (1)$

As given in the question,

$3 a + 2 b = 7 . . . E q (2)$

From both equations, we get $a = 1 & b = 2$

Then,

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

$\Rightarrow x = 1$

Vectors a i ^ + b j ^ + k ^ and 2 i ^ - 3 j ^ + 4 k ^ are perpendicular to each other when 3 a + 2 b = 7 ,…

Solution