Q75 / 150 · MHT CET 2026 (16 April Shift 2) · Mathematics

If $\bar{a} = 4\hat{i} + \hat{j} + \hat{k}$, $\bar{b} = 2\hat{i} + \hat{j} + 2\hat{k}$ and $\bar{c} = 3\hat{i} + 4\hat{j} + 5\hat{k}$, then $(\bar{a} + \bar{b}) \cdot (\bar{b} + \bar{c}) = $

  • A
    $30$
  • B
    $21$
  • Correct answer
    C
    $61$
  • D
    $10$

Solution

Given vectors are $\bar{a} = 4\hat{i} + \hat{j} + \hat{k}$, $\bar{b} = 2\hat{i} + \hat{j} + 2\hat{k}$ and $\bar{c} = 3\hat{i} + 4\hat{j} + 5\hat{k}$.

Calculating $\bar{a} + \bar{b}$:

$\bar{a} + \bar{b} = (4+2)\hat{i} + (1+1)\hat{j} + (1+2)\hat{k} = 6\hat{i} + 2\hat{j} + 3\hat{k}$

Calculating $\bar{b} + \bar{c}$:

$\bar{b} + \bar{c} = (2+3)\hat{i} + (1+4)\hat{j} + (2+5)\hat{k} = 5\hat{i} + 5\hat{j} + 7\hat{k}$

Taking the dot product:

$(\bar{a} + \bar{b}) \cdot (\bar{b} + \bar{c}) = (6\hat{i} + 2\hat{j} + 3\hat{k}) \cdot (5\hat{i} + 5\hat{j} + 7\hat{k})$

$= (6 \times 5) + (2 \times 5) + (3 \times 7)$

$= 30 + 10 + 21 = 61$

Answer: $61$

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

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