Question:
Let \(\vec c\) be a vector perpendicular to the vectors, \(\vec a=\hat i+\hat j-\hat k\) and \(\vec b=\hat i+2\hat j+\hat k\). If \(\vec c\cdot(\hat i+\hat j+3\hat k)=8\) then the value of \(\vec c\cdot(\vec a\times\vec b)\) is equal to __________.
Let \(\vec c\) be a vector perpendicular to the vectors, \(\vec a=\hat i+\hat j-\hat k\) and \(\vec b=\hat i+2\hat j+\hat k\). If \(\vec c\cdot(\hat i+\hat j+3\hat k)=8\) then the value of \(\vec c\cdot(\vec a\times\vec b)\) is equal to __________.
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A vector perpendicular to both \(\vec a\) and \(\vec b\) is \(\lambda(\vec a\times\vec b)\); find \(\lambda\) from the given dot product.
Updated On: Nov 4, 2026
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Correct Answer: 28
Solution and Explanation
\(\vec a\times\vec b=\begin{vmatrix}\hat i&\hat j&\hat k\\1&1&-1\\1&2&1\end{vmatrix}=\hat i(1+2)-\hat j(1+1)+\hat k(2-1)=3\hat i-2\hat j+\hat k\).
Since \(\vec c\perp\vec a,\vec b\), \(\vec c=\lambda(3\hat i-2\hat j+\hat k)\).
\(\vec c\cdot(\hat i+\hat j+3\hat k)=\lambda(3-2+3)=4\lambda=8\Rightarrow\lambda=2\).
\(\vec c\cdot(\vec a\times\vec b)=\lambda|\vec a\times\vec b|^2=2(9+4+1)=28\).
Answer: 28
Since \(\vec c\perp\vec a,\vec b\), \(\vec c=\lambda(3\hat i-2\hat j+\hat k)\).
\(\vec c\cdot(\hat i+\hat j+3\hat k)=\lambda(3-2+3)=4\lambda=8\Rightarrow\lambda=2\).
\(\vec c\cdot(\vec a\times\vec b)=\lambda|\vec a\times\vec b|^2=2(9+4+1)=28\).
Answer: 28
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