Question

find orthogonal vectors
suppose
\\(\vec{u} = 8\hat{i} + 5\hat{j} - 9\hat{k}\\),
\\(\vec{v} = 2\hat{i} - 4\hat{j} + 3\hat{k}\\),
\\(\vec{w} = \hat{i} + 2\hat{j} + 2\hat{k}\\).

Explanation:

Step1: Recall orthogonality condition

Two vectors are orthogonal if their dot product equals 0: $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 = 0$

Step2: Calculate $\vec{u} \cdot \vec{v}$

$\vec{u} \cdot \vec{v} = (8)(2) + (5)(-4) + (-9)(3) = 16 - 20 - 27 = -31
eq 0$

Step3: Calculate $\vec{u} \cdot \vec{w}$

$\vec{u} \cdot \vec{w} = (8)(1) + (5)(2) + (-9)(2) = 8 + 10 - 18 = 0$

Step4: Calculate $\vec{v} \cdot \vec{w}$

$\vec{v} \cdot \vec{w} = (2)(1) + (-4)(2) + (3)(2) = 2 - 8 + 6 = 0$

Answer:

The orthogonal vector pairs are $\vec{u}$ and $\vec{w}$, $\vec{v}$ and $\vec{w}$.