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Applications of Dot Product

A unit vector...

Question

A unit vector vectors perpendicular to each of the vectors $2^i -^j +^k$ and $3^i - 4^j -^k$ is

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Solution

$\begin{matrix} L e t t h e u n i t v e c t o r \to a = x ˆ i + y ˆ j + z ˆ k | a | = 1 = x^{2} + y^{2} + z^{2} - - - - (i) N o w, L e t \to b = 2 ˆ i - ˆ j + ˆ k \to c = 3 ˆ i - 4 ˆ j - ˆ k A / q \to a . \to b = (x ˆ i + y ˆ j + z ˆ k) (2 ˆ i - ˆ j + ˆ k) 2 x - y + z = 0 - - - - (i i) a n d, \to a . \to c = (x ˆ i + y ˆ j + z ˆ k) (3 ˆ i - 4 ˆ j - ˆ k) 3 x - 4 y - z = 0 - - - - (i i i) O n s o l v i n g e q u^{n} (i i) a n d (i i i) w e g e t x = \frac{1}{\sqrt{3}}, y = \frac{1}{\sqrt{3}} z = \frac{- 1}{\sqrt{3}} r e q u i r e d v e c t o r = \frac{1}{\sqrt{3}} (ˆ i + ˆ j - ˆ k) \end{matrix}$

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