Question:
Let $\vec{a}=\hat{i}+2\hat{j}+4\hat{k}$, $\vec{b}=2\hat{i}+4\hat{j}+8\hat{k}$ and $\vec{c}=2\hat{i}+4\hat{j}+3\hat{k}$. Then $(\vec{a}\times\vec{b})\cdot\vec{c}=$ ________.
Let $\vec{a}=\hat{i}+2\hat{j}+4\hat{k}$, $\vec{b}=2\hat{i}+4\hat{j}+8\hat{k}$ and $\vec{c}=2\hat{i}+4\hat{j}+3\hat{k}$. Then $(\vec{a}\times\vec{b})\cdot\vec{c}=$ ________.
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The scalar triple product is 0 if vectors are linearly dependent (parallel or coplanar).
Updated On: Jun 26, 2026
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Solution and Explanation
Step 1: Concept
The scalar triple product $(\vec{a}\times\vec{b})\cdot\vec{c}$ is zero if any two vectors are parallel.
Step 2: Meaning
Compare $\vec{a}$ and $\vec{b}$. $\vec{a} = \hat{i}+2\hat{j}+4\hat{k}$ and $\vec{b} = 2(\hat{i}+2\hat{j}+4\hat{k})$.
Step 3: Analysis
Since $\vec{b} = 2\vec{a}$, the vectors $\vec{a}$ and $\vec{b}$ are parallel, meaning $\vec{a}\times\vec{b} = \vec{0}$.
Step 4: Conclusion
$\vec{0}\cdot\vec{c} = 0$. Final Answer: (E)
The scalar triple product $(\vec{a}\times\vec{b})\cdot\vec{c}$ is zero if any two vectors are parallel.
Step 2: Meaning
Compare $\vec{a}$ and $\vec{b}$. $\vec{a} = \hat{i}+2\hat{j}+4\hat{k}$ and $\vec{b} = 2(\hat{i}+2\hat{j}+4\hat{k})$.
Step 3: Analysis
Since $\vec{b} = 2\vec{a}$, the vectors $\vec{a}$ and $\vec{b}$ are parallel, meaning $\vec{a}\times\vec{b} = \vec{0}$.
Step 4: Conclusion
$\vec{0}\cdot\vec{c} = 0$. Final Answer: (E)
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