# Let vectors a = 4i + 5j - k, b = i - 4j + 5k and c = 3i + j - k. Find a vector d which is perpendicular to both a and b and is such that d. c = 21.

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Let vectors a = 4i + 5j - k, b = i - 4j + 5k and c = 3i + j - k. Find a vector d which is perpendicular to both a and b and is such that d. c = 21.

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$\vec{a} =( \vec{4i} + \vec{5j} - \vec{k})$

$\vec{b} = (\vec{i} - \vec{4j} + \vec{5k})$

$\vec{c} = (\vec{3i} + \vec{j} - \vec{k})$

Let   $\vec{d} = \vec{pi} + \vec{qj} - \vec{rk}$

the vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$

Solving equations 1,2,3 simultaneously we get

p = 7,q = - 7,r = - 7