Q73 / 160 · AP EAMCET 2019 (20 Apr Shift 2) · Mathematics

Let $\mathbf{a}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{c}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$. The volume (in cubic units) of the parallelopiped having $\mathbf{a}+\mathbf{b}+\mathbf{c}, \mathbf{a}-\mathbf{b}+\mathbf{c}$ and $\mathbf{a}+\mathbf{b}-\mathbf{c}$ as coterminus edges is

  • A
    6
  • B
    7
  • Correct answer
    C
    28
  • D
    36

Solution

It is given that, $$ \mathbf{a}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}} $$ and $\mathbf{c}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ So, vectors $$ \begin{aligned} \mathbf{a}+\mathbf{b}+\mathbf{c} & =6 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}} \\ \mathbf{a}-\mathbf{b}+\mathbf{c} & =4 \hat{\mathbf{i}}-6 \hat{\mathbf{j}}+7 \hat{\mathbf{k}} \\ \text { and } \quad \mathbf{a}+\mathbf{b}-\mathbf{c} & =0 \hat{\mathbf{i}}-0 \hat{\mathbf{j}}+\hat{\mathbf{k}} \end{aligned} $$ So, the required volume of the parallelopiped having $\mathbf{a}+\mathbf{b}+\mathbf{c}, \mathbf{a}-\mathbf{b}+\mathbf{c}$ and $\mathbf{a}+\mathbf{b}-\mathbf{c}$ as coterminus edges is $$ v=\left\|\begin{array}{ccc} 6 & -2 & 3 \\ 4 & -6 & 7 \\ 0 & 0 & 1 \end{array}\right\|=|1(-36+8)|=|-28|=28 \text {. } $$ Hence, option (c) is correct.

Hence, the correct answer is option: 'C'.

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