QuestionGiven $\mathbf{u}_{1}=(6,-1)$ and $\mathbf{u}_{2}=(3,2)$, if we let $\mathbf{v}_{1}=\mathbf{u}_{1}$, use the Gram-Schmidt process to find $\mathbf{v}_{2}$ If needed, enter your answers as fractions, not decimals.
This question accepts'answers that are in a form like " $(-1,3)$ " or " $(3,7,3 z)$ ". The entries can be numbers or formulas. Help | Preview

Studdy Solution
Subtract the projection from $\mathbf{u}_2$ to find $\mathbf{v}_2$:
$\mathbf{v}_2 = \mathbf{u}_2 - \text{proj}_{\mathbf{v}_1} \mathbf{u}_2 = (3, 2) - \left(\frac{96}{37}, -\frac{16}{37}\right)$
Calculate each component:
$\mathbf{v}_2 = \left(3 - \frac{96}{37}, 2 + \frac{16}{37}\right)$
Simplify the components:
$\mathbf{v}_2 = \left(\frac{111}{37} - \frac{96}{37}, \frac{74}{37} + \frac{16}{37}\right) = \left(\frac{15}{37}, \frac{90}{37}\right)$
The orthogonal vector $\mathbf{v}_2$ is:
$\boxed{\left(\frac{15}{37}, \frac{90}{37}\right)}$