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Applications of Dot Product

If a=i+2j-3k ...

Question

If $a = i + 2 j - 3 k$ and $b = 3 i - j + 2 k$ , then the angle between the vectors $a + b$ and $a - b$ is?

A

$60 °$

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B

$90 °$

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C

$45 °$

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D

$55 °$

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Solution

The correct option is B
$90 °$

Finding the angle:
Given,
$a = i + 2 j - 3 k$ & $b = 3 i - j + 2 k$
$a + b = (i + 2 j - 3 k) + (3 i - j + 2 k) a + b = 4 i + j - k$
Again,
$a - b = (i + 2 j - 3 k) - (3 i - j + 2 k) a - b = - 2 i + 3 j - 5 k$
Now,
$\begin{array}{rcl} (a + b) . (a - b) & = & (4 i + j - k) . (- 2 i + 3 j - 5 k) \\ = & - 8 + 3 + 5 \\ = & 0 \end{array}$
Since, the dot product of two vector $(a + b) & (a - b)$ is $0$ .
Therefore, they are perpendicular to each other & the angle between them is $90 °$
Hence, correct option is $(B)$

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