Subject: Applied Mathematics 4

Topic: Vector Spaces

Difficulty: Medium

Let, $ w = (x_1,x_2,x_3) $ be orthogonal to both $ \bar{u} $ and $ \bar{v} $

Here, $ \bar{u} = -6 \bar{i} + 4 \bar{j} + 2\bar{k} \\ \bar{v} = 3 \bar{i} + \bar{j} + 5 \bar{k} \\ \bar{w} = x_1 \bar{i} + x_2 \bar{j} + x_3 \bar{k} $

Since $ \bar{w}$ is orthogonal to both $ \bar{u} $ and $ \bar{v} $,

$ \bar{u}.\bar{w} = 0 \\ -6x_1 + 4x_2 + 2x_3 = 0 $

Also,

$ \bar{v}.\bar{w} = 0 \\ 3x_1 + x_2 + 5x_3 = 0 $

Applying Crammers rule,

$ \frac{x_1}{18} = \frac{- x_2}{-36} = \frac{x_3}{-18} \\ \frac{x_1}{1} = \frac{x_2}{2} = \frac{x_3}{-1} \\ w = (1,2,-1) $

Hence, w = (1,2,-1) is orthogonal to both $ \bar{u} $ and $ \bar{v} $.