if \(\vec{a}\)= \(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\), \(\vec{b}\)= 2\(\hat{i}\) -\(\hat{j}\)+ 3\(\hat{k}\) and \(\vec{c}\) = \(\hat{i}\) - 2 \(\hat{j}\) + \(\hat{k}\) then, find a unit vector parallel to the vectors 2\(\vec{a}\) -\(\vec{b}\)+ 3\(\vec{c}\)
Answer & explanation
Correct answer: option 1
We have \(\vec{a}\)= \(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\), \(\vec{b}\)= 2\(\hat{i}\) -\(\hat{j}\)+ 3\(\hat{k}\) and \(\vec{c}\) = \(\hat{i}\) - 2 \(\hat{j}\) + \(\hat{k}\)
then 2\(\vec{a}\) -\(\vec{b}\)+ 3\(\vec{c}\) = 2( \(\hat{i}\)+ \(\hat{j}\)+ \(\hat{k}\)) - (2\(\hat{i}\)- \(\hat{j}\) ̂+ 3\(\hat{k}\)) + 3(\(\hat{i}\)- 2 \(\hat{j}\)+ \(\hat{k}\))
= (3\(\hat{i}\)- 3\(\hat{j}\)+ 2\(\hat{k}\))
|2\(\vec{a}\) -\(\vec{b}\)+ 3\(\vec{c}\)| = √(3)2 + (-3)2 +(2)2 = √22
Hence, the unit vector along 2\(\vec{a}\) -\(\vec{b}\)+ 3\(\vec{c}\) is
= (2\(\vec{a}\) -\(\vec{b}\)+ 3\(\vec{c}\))/|2\(\vec{a}\) -\(\vec{b}\)+ 3\(\vec{c}\)| = (3\(\hat{i}\)- 3\(\hat{j}\)+ 2\(\hat{k}\))/ √22