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Applications of Dot Product

Calculate the...

Question

Calculate the scalar product of the following vectors.
Find the vector c being given that it is perpendicular to the vectors $a = {2, 3, - 1}, b = {1, - 2, 3}$ and satisfies the condition $c . {2 i - j + k} = - 6.$

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Solution

Let c be a vector (x,y,z)
According to given conditions
$a . c = 2 x + 3 y - z = 0................. (1)$
$b . c = x - 2 y + 3 z = 0.................. (2)$
$c . (2 \to i - \to j + \to k) = 2 x - y + z = - 6............ (3)$
Equations $(1) - 2 \times (2)$
$\Rightarrow y = z . . . . . . . . . (4)$
Equations $3 \times (1) + (2)$
$\Rightarrow x = - y . . . . . (5)$
Substitute Equation $4 a n d 5 i n 3$
$- 2 y - y + y = - 6$
$y = 3$
Hence $\to c = (- 3, 3, 3)$

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