Question

Let u,v1 and v2 be vectors in R^3, and let c1 and c2 be scalars. If u is orthogonal to both v1 and v2, prove that u is orthogonal to the vector c1v1+c2v2.

Vectors and spaces
ANSWERED
asked 2021-03-02

Let u,\(v_1\) and \(v_2\) be vectors in \(R^3\), and let \(c_1\) and \(c_2\) be scalars. If u is orthogonal to both \(v_1\) and \(v_2\), prove that u is orthogonal to the vector \(c_1v_1+c_2v_2\).

Answers (1)

2021-03-03

Since u is orthogonal to \(v_1\) and \(v_2\), we have that \(u\times v_1=0\)
\(u\times v_2=0\)
Now, \(u\times(c_1v_1+c_2v_2)=u\times(c_1v)+u(c_2v_2) =c_1u\times v_1+c_2u\times v_2 =c_1\times 0+c_2\times0 =0\)
Therefore, u is orthogonal to \(c_1v_1+c_2v_2\), as required.

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