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$$\mathbf u = \begin{pmatrix} 2 \\ 14 \\ -4 \\ 1 \end{pmatrix},\mathbf{v_1} = \begin{pmatrix} 1/\sqrt{5} \\ 2/\sqrt{5} \\ 0 \\ 0 \end{pmatrix}, \mathbf{v_2} = \begin{pmatrix} 2/\sqrt{30} \\ -1/\sqrt{30} \\ 5/\sqrt{30} \\ 0 \end{pmatrix}$$

Have to find the component of $\mathbf u$ orthogonal to both $\mathbf{v_1}$ and $\mathbf{v_2}$. I know how to do this if I just had to find the component of $\mathbf u$ orthogonal to one vector using the equation: $$ u- \frac {u \cdot v}{ v \cdot v}v $$

But I'm not sure how to do it with two vectors, any help is appreciated.

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You should take a look at the Gram-Schmidt algorithm. Note that $v_1,v_2$ are orthogonal vectors. $$ u_1=u-{u\cdot v_1\over v_1\cdot v_1}v_1=\left[-4\,2\,-4\,1\right]^\top\\ u_2=u_1-{u_1\cdot v_2\over v_2\cdot v_2}v_2=\left[-2\,1\,1\,1\right]^\top\\ $$

$u_j$ is orthogonal to $v_j$ by your formula and $u_2$ is orthogonal to $v_1$ as $$ u_2\cdot v_1=u_1\cdot v_1-{u_1\cdot v_2\over v_2\cdot v_2}v_2\cdot v_1=0-{u_1\cdot v_2\over v_2\cdot v_2}0=0 $$ So $u_2$ is orthogonal to $v_1,v_2$.

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