if a= i+2j+k , b= i-j+k and c= i+j-k. Fina a vector in the plane of a and b whose projection on c is 1/under root 3.

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Solution

$a vector to be in the plane of a = i + 2 j + k and b = i - 2 j + k must have z component as + 1 so vector will be in the form of xi + yj + k its projetion on c vector i + j - k using formula projection of \vec{a} on \vec{b} = \frac{\vec{a} . \vec{b}}{\vec{b}} = \frac{(x i + y j + k) . (i + j - k)}{\sqrt{3}} = \frac{1}{\sqrt{3}} x + y - 1 = 1 x + y = 2 so any vector whose which satisfy x + y = 2 and z = + 1 will be in the plane of a and b and its projection on c will be \frac{1}{\sqrt{3}} ans : Required vector is i + j + k$

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