Find a vector of magnitude 5 units, and parallel to the resultant of vectors \(\vec{a}\) = 2\(\hat{i}\) +3\(\hat{j}\) - \(\hat{k}\) and \(\vec{b}\) = \(\hat{i}\) -2\(\hat{j}\) + \(\hat{k}\)
Answer & explanation
Correct answer: option 4
We have \(\vec{a}\) = 2\(\hat{i}\) +3\(\hat{j}\) - \(\hat{k}\) and \(\vec{b}\) = \(\hat{i}\) -2\(\hat{j}\) + \(\hat{k}\)
let \(\vec{c}\) be the resultant of \(\vec{a}\) and \(\vec{b}\)
Then,
\(\vec{c}\) = \(\vec{a}\) + \(\vec{b}\) =(2\(\hat{i}\) +3\(\hat{j}\)- \(\hat{k}\)) + ( \(\hat{i}\) -2\(\hat{j}\) + \(\hat{k}\)) = 3 \(\hat{i}\) +\(\hat{j}\)
|\(\vec{c}\)| = √(3)2 +(1)2
|\(\vec{c}\)|= √10
so unit vector \(\hat{c}\) = \(\vec{c}\) / |\(\vec{c}\)| = (3 \(\hat{i}\)+\(\hat{j}\) ̂)/ √10
Hence vector of magnitude 4 units, and parallel to the resultant of vectors \(\vec{a}\) and \(\vec{b}\) is = ±5. {(3 \(\hat{i}\) + \(\hat{j}\))/ √(10)