What is the condition for the vectors $2 i + 3 j - 4 k$ and $3 i - a j + b k$ to be parallel?

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Solution

Determine the condition for the vectors $2 i + 3 j - 4 k$ and $3 i - a j + b k$ to be parallel.
When two vectors are parallel to each other, the coefficients i, j, and k must have the same ratio in both vectors since we must have the same direction for both vectors.
$a_{1} \hat{i} + b_{1} \hat{j} + c_{1} \hat{k} a n d a_{2} \hat{i} + b_{2} \hat{j} + c_{2} \hat{k}$
Now, consider the parallel condition of two vectors.
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
so we have $2 \hat{i} + 3 \hat{j} - 4 \hat{k} a n d 3 \hat{i} - a \hat{j} + b \hat{k}$ Now by the above condition $\frac{2}{3} = \frac{3}{- a} = \frac{- 4}{b}$ so we have $a = - 4.5$ and $b = - 6$ .
Hence, the condition for the vectors $2 i + 3 j - 4 k$ and $3 i - a j + b k$ to be parallel, When two given vectors are parallel, we get $a = - 4.5$ and $b = - 6$ .

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