Question

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 $3 a+2 b=7$, the ratio of $a$ to $b$ is $\frac{x}{2}$ The value of $x$ is __

Answer 1

Correct Answer: 1

For two vectors to be perpendicular, their dot product must be zero: \[ \vec{a} \cdot \vec{b} = 0. \] Substitute \(\vec{a} = a\hat{i} + b\hat{j} + \hat{k}\) and \(\vec{b} = 2\hat{i} - 3\hat{j} + 4\hat{k}\): \[ (a\hat{i} + b\hat{j} + \hat{k}) \cdot (2\hat{i} - 3\hat{j} + 4\hat{k}) = 0. \] Simplify: \[ 2a - 3b + 4 = 0. \] This gives the first equation: \[ 2a - 3b = -4. \] From the problem, another equation is given: \[ 3a + 2b = 7. \] Solve the simultaneous equations: Multiply equation (1) by 2: \[ 4a - 6b = -8. \] Multiply equation (2) by 3: \[ 9a + 6b = 21. \] Add equations (3) and (4): \[ 13a = 13 \quad \Rightarrow \quad a = 1. \] Substitute \(a = 1\) into equation (2): \[ 3(1) + 2b = 7 \quad \Rightarrow \quad 3 + 2b = 7 \quad \Rightarrow \quad 2b = 4 \quad \Rightarrow \quad b = 2. \] The ratio of \(a\) to \(b\) is: \[ \frac{a}{b} = \frac{x}{2} \quad \Rightarrow \quad \frac{1}{2} = \frac{x}{2}. \] Thus: \[ x = 1. \]