Answer:
The values of [tex]m[/tex] and [tex]n[/tex] are 0 and 2, respectively.
Step-by-step explanation:
If [tex]\vec {c} = m\,\hat{i} + n\,\hat{j} + \hat{k}[/tex] is perpendicular to [tex]\vec {a} = 5\,\hat{i} + \hat{j} -2\,\hat{k}[/tex] and [tex]\vec {b} = 3\,\hat{i} - 3\,\hat{j} + 6\,\hat{k}[/tex], then the following relationships must be observed:
[tex]\vec {c}\,\bullet\,\vec {a} = 0[/tex] (1)
[tex]\vec{c}\,\bullet \,\vec{a} = 0[/tex] (2)
Then, we expand the previous expressions:
[tex](m, n, 1)\,\bullet\,(5, 1, -2) = 0[/tex]
[tex]5\cdot m + n = 2[/tex] (1b)
[tex](m, n, 1)\,\bullet\,(3, -3, 6) = 0[/tex]
[tex]3\cdot m - 3\cdot n = -6[/tex] (2b)
Then, we solve for [tex]m[/tex] and [tex]n[/tex]:
[tex]m = 0, n = 2[/tex]
The values of [tex]m[/tex] and [tex]n[/tex] are 0 and 2, respectively.
