Question:
If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0}, \] then the values of \(p\) and \(q\) are:
If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0}, \] then the values of \(p\) and \(q\) are:
Show Hint
Avoid calculating the full determinant for the cross product if it equals zero; the proportionality rule is much faster.
Always keep track of negative signs in component ratios.
Always keep track of negative signs in component ratios.
Updated On: Sep 11, 2026
- \(p = -\frac{2}{3}, q = \frac{5}{3}\)
- \(p = -\frac{8}{3}, q = \frac{20}{3}\)
- \(p = \frac{20}{3}, q = -\frac{8}{3}\)
- \(p = 0, q = 0\)
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The Correct Option is B
Solution and Explanation
Concept:
• The cross product of two non-zero vectors is the zero vector (\(\vec{0}\)) if and only if the vectors are parallel or collinear.
• If two vectors \(\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\) and \(\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}\) are parallel, their corresponding components are proportional: \(\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}\).
Step 1: Apply the condition for parallel vectors
Given \(\vec{a} = 3\hat{i} - 2\hat{j} + 5\hat{k}\) and \(\vec{b} = 4\hat{i} + p\hat{j} + q\hat{k}\).
Since \(\vec{a} \times \vec{b} = \vec{0}\), the vectors are parallel.
Therefore, the ratios of their components must be equal:
\[ \frac{3}{4} = \frac{-2}{p} = \frac{5}{q} \]
Step 2: Solve for p
Equate the first and second ratios:
\[ \frac{3}{4} = \frac{-2}{p} \] \[ 3p = -8 \] \[ p = -\frac{8}{3} \]
Step 3: Solve for q
Equate the first and third ratios:
\[ \frac{3}{4} = \frac{5}{q} \] \[ 3q = 20 \] \[ q = \frac{20}{3} \] Thus, the values are \(p = -\frac{8}{3}\) and \(q = \frac{20}{3}\).
• The cross product of two non-zero vectors is the zero vector (\(\vec{0}\)) if and only if the vectors are parallel or collinear.
• If two vectors \(\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\) and \(\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}\) are parallel, their corresponding components are proportional: \(\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}\).
Step 1: Apply the condition for parallel vectors
Given \(\vec{a} = 3\hat{i} - 2\hat{j} + 5\hat{k}\) and \(\vec{b} = 4\hat{i} + p\hat{j} + q\hat{k}\).
Since \(\vec{a} \times \vec{b} = \vec{0}\), the vectors are parallel.
Therefore, the ratios of their components must be equal:
\[ \frac{3}{4} = \frac{-2}{p} = \frac{5}{q} \]
Step 2: Solve for p
Equate the first and second ratios:
\[ \frac{3}{4} = \frac{-2}{p} \] \[ 3p = -8 \] \[ p = -\frac{8}{3} \]
Step 3: Solve for q
Equate the first and third ratios:
\[ \frac{3}{4} = \frac{5}{q} \] \[ 3q = 20 \] \[ q = \frac{20}{3} \] Thus, the values are \(p = -\frac{8}{3}\) and \(q = \frac{20}{3}\).
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