Question:
If \(\overset{̄}{a} = \hat{i}+\hat{j}+\hat{k},\overset{̄}{b} = \hat{i},\overset{̄}{c} = c_1\hat{i}+c_2\hat{j}+c_3\hat{k}\) with \(c_1 = 1\), \(c_2 = 2\), then value of \(c_3\) such that \(\overset{̄}{a},\overset{̄}{b},\overset{̄}{c}\) are coplanar is ____
If \(\overset{̄}{a} = \hat{i}+\hat{j}+\hat{k},\overset{̄}{b} = \hat{i},\overset{̄}{c} = c_1\hat{i}+c_2\hat{j}+c_3\hat{k}\) with \(c_1 = 1\), \(c_2 = 2\), then value of \(c_3\) such that \(\overset{̄}{a},\overset{̄}{b},\overset{̄}{c}\) are coplanar is ____
Show Hint
Three vectors are coplanar when their scalar triple product (determinant) is zero.
Updated On: Oct 1, 2026
- \(2\)
- \(-1\)
- \(0\)
- \(-2\)
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Key Formula:
Vectors \(\vec a,\vec b,\vec c\) are coplanar if \(\begin{vmatrix}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{vmatrix} = 0\).
Step 2: Set up:
\(\vec a = (1,1,1)\), \(\vec b = (1,0,0)\), \(\vec c = (1,2,c_3)\).
Step 3: Expand:
Expand along the second row: \(-1\cdot\begin{vmatrix}1&1\\2&c_3\end{vmatrix} = -(c_3 - 2) = 2 - c_3\).
Setting the determinant to zero gives \(c_3 = 2\).
Final Answer:
The value of \(c_3\) is \(2\), option (A). \[ \boxed{2} \]
Vectors \(\vec a,\vec b,\vec c\) are coplanar if \(\begin{vmatrix}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{vmatrix} = 0\).
Step 2: Set up:
\(\vec a = (1,1,1)\), \(\vec b = (1,0,0)\), \(\vec c = (1,2,c_3)\).
Step 3: Expand:
Expand along the second row: \(-1\cdot\begin{vmatrix}1&1\\2&c_3\end{vmatrix} = -(c_3 - 2) = 2 - c_3\).
Setting the determinant to zero gives \(c_3 = 2\).
Final Answer:
The value of \(c_3\) is \(2\), option (A). \[ \boxed{2} \]
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