a→=2i^−j^+k^,b→=i^+2j^−k^,c→=i^+j^−2k^.  A vector co-planar with  b→ and c→ whose projection on a→  is magnitude  23 is

Let the required vector be r→ thenr→=x1b→+x2c→ and r→⋅a→=23(|a→|)=2 Now,  r→⋅a→=x1a→⋅b→+x2a→⋅c→⇒ 2=x1(2−2−1)+x2(2−1−2)⇒ x1+x2=−2⇒ r→=x1(i^+2j^−k^)+x2(i^+j^−2k^)=i^x1+x2+j^2x1+x2−k^2x2+x1=−2i^+j^x1−2−k^−4−x1