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NDA 1 2024 Maths, question 69

Topic: Vectors

Q69

For any vector r→\overrightarrow{r}, what is (r→⋅i^)(r→×i^)+(r→⋅j^)(r→×j^)+(r→⋅k^)(r→×k^)(\overrightarrow{r}\cdot \hat{i})(\overrightarrow{r}\times \hat{i})+(\overrightarrow{r}\cdot \hat{j})(\overrightarrow{r}\times \hat{j})+(\overrightarrow{r}\cdot \hat{k})(\overrightarrow{r}\times \hat{k}) equal to ?

  1. (a) 0→\overrightarrow{0}Correct answer
  2. (b) r→\overrightarrow{r}
  3. (c) 2r→2\overrightarrow{r}
  4. (d) 3r→3\overrightarrow{r}

Solution.

Answer: (a) 0→\overrightarrow{0}

  1. The coefficients r·i, r·j and r·k are the Cartesian components of r.
  2. By distributivity, the displayed sum is r × [(r·i)i + (r·j)j + (r·k)k] = r × r = 0.