Find the values of 'a' for which the vectors \[\vec{\alpha} = \hat {i} + 2 \hat {j} + \hat {k} , \vec{\beta} = a \hat {i} + \hat {j} + 2 \hat {k} \text { and } \vec{\gamma} = \hat {i} + 2 \hat {j} + a \hat { k }\] are coplanar.

Given: \[ \vec{\alpha} = \hat {i} + 2 \hat {j} + \hat {k} \] \[ \vec{\beta} = a \hat {i} + \hat {j} + 2 \hat {k} \] \[ \vec{\gamma} = \hat {i} + 2 \hat {j} + a \hat {k}\] \[\text {We know that three vectors } \vec{\alpha} , \vec{\beta} , \vec{\gamma} \text { are coplanar iff their scalar product is zero } . \] \[ \therefore \left[ \vec{\alpha} \vec{\beta} \vec{\gamma} \right] = 0\] \[ \Rightarrow \begin{vmatrix}1 & 2 & 1 \\ a & 1 & 2 \\ 1 & 2 & a\end{vmatrix} = 0\] \[ \Rightarrow 1\left( a - 4 \right) - 2\left( a^2 - 2 \right) + 1\left( 2a - 1 \right) = 0\] \[ \Rightarrow - 2 a^2 + 3a - 1 = 0\] \[ \Rightarrow 2 a^2 - 3a + 1 = 0\] \[ \Rightarrow \left( a - 1 \right)\left( 2a - 1 \right) = 0\] \[ \Rightarrow a = 1, \frac{1}{2}\]

Find the values of 'a' for which the vectors → α = ^ i + 2 ^ j + ^ k , → β = a ^ i + ^ j + 2 ^ k and → γ = ^ i + 2 ^ j + a ^ k are coplanar. - Mathematics

प्रश्न

Find the values of 'a' for which the vectors

$\vec{α} = \hat{i} + 2 \hat{j} + \hat{k}, \vec{β} = a \hat{i} + \hat{j} + 2 \hat{k} and \vec{γ} = \hat{i} + 2 \hat{j} + a \hat{k}$ are coplanar.

योग

उत्तर

Given:

$\vec{α} = \hat{i} + 2 \hat{j} + \hat{k}$

$\vec{β} = a \hat{i} + \hat{j} + 2 \hat{k}$

$\vec{γ} = \hat{i} + 2 \hat{j} + a \hat{k}$

$We know that three vectors \vec{α}, \vec{β}, \vec{γ} are coplanar iff their scalar product is zero .$

$∴ [\vec{α} \vec{β} \vec{γ}] = 0$

$\Rightarrow | \begin{matrix} 1 & 2 & 1 \\ a & 1 & 2 \\ 1 & 2 & a \end{matrix} | = 0$

$\Rightarrow 1 (a - 4) - 2 (a^{2} - 2) + 1 (2 a - 1) = 0$

$\Rightarrow - 2 a^{2} + 3 a - 1 = 0$

$\Rightarrow 2 a^{2} - 3 a + 1 = 0$

$\Rightarrow (a - 1) (2 a - 1) = 0$

$\Rightarrow a = 1, \frac{1}{2}$

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Scalar Triple Product of Vectors

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Q 4 Q 3 Q 5

अध्याय 26: Scalar Triple Product - Exercise 26.1 [पृष्ठ १७]

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आरडी शर्मा Mathematics [English] Class 12

अध्याय 26 Scalar Triple Product
Exercise 26.1 | Q 4 | पृष्ठ १७

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Scalar Triple Product of Vectors

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