If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are coterminous edges of $\mathbf{a}$ parallelopiped such that…

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are coterminous edges of $\mathbf{a}$ parallelopiped such that $\left|\begin{array}{lll}\mathbf{a} \cdot \mathbf{a} & \mathbf{a} \cdot \mathbf{b} & \mathbf{a} \cdot \mathbf{c} \\ \mathbf{b} \cdot \mathbf{a} & \mathbf{b} \cdot \mathbf{b} & \mathbf{b} \cdot \mathbf{c} \\ \mathbf{c} \cdot \mathbf{a} & \mathbf{c} \cdot \mathbf{b} & \mathbf{c} \cdot \mathbf{c}\end{array}\right|=16$, then find the volume of the parallelopiped
  1. 32
  2. 16
  3. 4
  4. 8

Solution

It is given that, $\left|\begin{array}{lll}\mathbf{a} \cdot \mathbf{a} & \mathbf{a} \cdot \mathbf{b} & \mathbf{a} \cdot \mathbf{c} \\ \mathbf{b} \cdot \mathbf{a} & \mathbf{b} \cdot \mathbf{b} & \mathbf{b} \cdot \mathbf{c} \\ \mathbf{c} \cdot \mathbf{a} & \mathbf{c} \cdot \mathbf{b} & \mathbf{c} \cdot \mathbf{c}\end{array}\right|=16$ Let $\mathbf{a}=a_1 \hat{\mathbf{I}}+a_2 \hat{\mathbf{J}}+a_3 \hat{\mathbf{k}}, \mathbf{b}=b_1 \hat{\mathbf{I}}+b_2 \hat{\mathbf{J}}+b_3 \hat{\mathbf{k}}$ and $\mathbf{c}=c_1 \hat{\mathbf{i}}+c_2 \hat{\mathbf{j}}+c_3 \hat{\mathbf{k}}$ $ \begin{aligned} & \therefore\left|\begin{array}{lll} \mathbf{a} \cdot \mathbf{a} & \mathbf{a} \cdot \mathbf{b} & \mathbf{a} \cdot \mathbf{c} \\ \mathbf{b} \cdot \mathbf{a} & \mathbf{b} \cdot \mathbf{b} & \mathbf{b} \cdot \mathbf{c} \\ \mathbf{c} \cdot \mathbf{a} & \mathbf{c} \cdot \mathbf{b} & \mathbf{c} \cdot \mathbf{c} \end{array}\right| \\ & =\left\lvert\, \begin{array}{cc} a_1^2+a_2^2+a_3^2 & a_1 b_1+a_2 b_2+a_3 b_3 \\ b_1 a_1+b_2 a_2+b_3 a_3 & b_1^2+b_2^2+b_3^2 \\ c_1 a_1+c_2 a_2+c_3 a_3 & c_1 b_1+c_2 b_2+c_3 b_3 \end{array}\right. \\ & a_1 c_1+a_2 c_2+a_3 c_3 \\ & b_1 c_1+b_2 c_2+b_3 c_3 \\ & c_1^2+c_2^2+c_3^2 \\ & \end{aligned} $ $ \begin{aligned} & =\left|\begin{array}{lll} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{array}\right|\left|\begin{array}{lll} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{array}\right| \\ & =\left[\begin{array}{lll} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{array}\right]\left[\begin{array}{lll} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{array}\right]=16 \\ & \therefore \quad\left[\begin{array}{lll} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{array}\right]=4 \\ & \end{aligned} $ (given) So, the volume of the parallelopiped having coterminous edges is 4

Asked in: AP EAMCET 2020 (22 Sep Shift 1)

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