Q263 / 270 · NDA 2020 · Mathematics

Two adjacent sides of a parallelogram are $2 \hat{i}-4 \hat{j}+5 \hat{k}$ and $\hat{i}-2 \hat{j}-3 \hat{k}$. What is the magnitude of dot product of vectors which represent its diagonals?

  • A
    21
  • B
    25
  • Correct answer
    C
    31
  • D
    36

Solution

Let $\vec{a}=2 \hat{i}-4 \hat{j}+5 \hat{k}$ and $\vec{b}=\hat{i}-2 \hat{j}-3 \hat{k}$ Two diagonals of the parallelograms are given by $\begin{aligned} (\vec{a}+\vec{b}) & =(2+1) \hat{i}-(4+2) \hat{j}+(5-3) \hat{k} \\ & =3 \hat{i}-6 \hat{j}+2 \hat{k} \end{aligned}$ and $\begin{aligned} (\vec{a}-\vec{b}) & =(2-1) \hat{i}-(4-2) \hat{j}+(5-(-3)) \hat{k} \\ & =\hat{i}-2 \hat{j}+8 \hat{k} \end{aligned}$ Dot products of the diagonals $\begin{gathered}(\vec{a}+\vec{b}) \cdot(\vec{a}-\vec{b})=(3 \hat{i}-6 \hat{j}+2 \hat{k}) \cdot(\hat{i}-2 \hat{j}+8 \hat{k}) \\ =(3+12+16)=31 \text { units }\end{gathered}$

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

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