Vector Algebra
Vector Algebra
nta_abhyas_2025
Grade 12
Question:
If $\begin{vmatrix} \vec{a} - 2\vec{b} + \vec{c} & \vec{b} + 2\vec{c} + 3\vec{a} \\ \vec{a} & \vec{b} & \vec{c} \\ \vec{b} & \vec{b} & \vec{c} \\ \vec{c} & \vec{c} & \vec{c} \end{vmatrix} = 54$, where $\vec{a}$, $\vec{b}$, $\vec{c}$ are 3 non-coplanar vectors, then the value of $[\vec{a} \, \vec{b} \, \vec{c}]$ is equal to
Step-by-Step Solution
Key Concept: Finding magnitude of a position vector subject to constraint equations requires substituting relationships and computing the norm
Given $\vec{r} = 2\vec{i} - \vec{j} - 4\vec{k}$ and $\vec{r}$ with $\vec{x} = 2y - 4z$ and $x = 8$. Let $\vec{r} = x\vec{i} + y\vec{j} + z\vec{k}$, then $x = 2y - 4z$ and $x = 8$, giving $2y - 4z = 8$. The magnitude is $|\vec{r}| = \sqrt{64 + (4z)^2 + (2z)^2 + z^2} = \sqrt{64 + 4 + 4 + 1} = \sqrt{84}$ when the constraint is satisfied. With $x = 8$ and solving for integer values: if $2y - 4z = 8$, then $y - 2z = 4$. Testing $z = 2$ gives $y = 4 + 4 = 8$, but using the given form $\vec{r} = 8\vec{i} + 4\vec{j} + 2\vec{k}$ yields $|\vec{r}| = \sqrt{64 + 16 + 4} = \sqrt{84} = 2\sqrt{21}$.
Correct Answer: D