If a = i - 2 j + 5 k and b = 2 i + j - 3 k then what is ≤ft( { b - a} ) · ≤ft( {3 a + b} ) equal to?

Correct Answer is : −106-106−106

Correct Answer is : −106-106−106

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UPSC NDA I 2019 Mathematics Solved Paper

Question : 66 of 120

Marks: +1, -0

\vec a = \hat i - 2\hat j + 5\hat k

\vec b = 2\hat i + \hat j - 3\hat k

\left( {\vec b - \vec a} \right) \cdot \left( {3\vec a + \vec b} \right)

equal to?

Solution:

Concept:

The dot product of two vectors is the sum of the products of their corresponding components.

Scalar multiplication multiplies each component of the vector by the scalar.

Explanation:

Step 1: Compute

\vec b - \vec a

\vec b - \vec a = (2\hat i + \hat j - 3\hat k) - (\hat i - 2\hat j + 5\hat k) = (2-1)\hat i + (1+2)\hat j + (-3-5)\hat k = \hat i + 3\hat j - 8\hat k

Step 2: Compute

3\vec a + \vec b

First,

3\vec a = 3(\hat i - 2\hat j + 5\hat k) = 3\hat i - 6\hat j + 15\hat k

Then,

3\vec a + \vec b = (3\hat i - 6\hat j + 15\hat k) + (2\hat i + \hat j - 3\hat k) = (3+2)\hat i + (-6+1)\hat j + (15-3)\hat k = 5\hat i - 5\hat j + 12\hat k

Step 3: Compute the dot product

(\vec b - \vec a) \cdot (3\vec a + \vec b)

(\hat i + 3\hat j - 8\hat k) \cdot (5\hat i - 5\hat j + 12\hat k) = (1)(5) + (3)(-5) + (-8)(12) = 5 - 15 - 96 = -106

Answer:

The value is

-106

, which corresponds to option B.

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