Question

Two vectors are given by $A = \left( {4.0m} \right)\hat i - \left( {3.0m} \right)\hat j + \left( {1.0m} \right)\hat k$ and $B = \left( { - 1.0m} \right)\hat i + \left( {1.0m} \right)\hat j + \left( {4.0m} \right)\hat k$. How to find a third vector C such that $A - B + C = 0?$

Answer 1

Hint:In the respective question, we are provided with the two vectors. And we have to find the third vector which satisfies the given condition which is A-B+C=0. Take the C on the left-hand side and put the given values. After using the law of subtraction of vectors we can find our answer.

Complete step by step answer:
According to the question, we are provided with two vectors. Vectors are the quantities which have magnitude as well as direction. These are represented by arrows which are not shown in the question. But for our understanding we can apply an arrow on the top of the vectors as $\vec A$ . Vectors are represented as $\hat i,\hat j,\hat k$ which represents x, y and z axis respectively.

Now, in the question we have to find the third vector which satisfies the relation given which is $A - B + C = 0$ which can also be written as $C = B - A$ because we have to find the third vector C.
Putting the values of the vectors and using the law of subtraction of vectors.
$C = \left[ { - 1m\hat i + 1m\hat j + 4m\hat k} \right] - \left[ {4m\hat i - 3m\hat j + 1m\hat k} \right]$
On solving,
$C = \left( { - 1m - 4m} \right)\hat i + \left( {1m + 3m} \right)\hat j + \left( {4m - 1m} \right)\hat k$
$\therefore C = - 5m\hat i + 4m\hat j + 3m\hat k$
Which is our required third vector.

Note:Subtraction of vectors is the same as the addition of the vectors; the only difference is just by reversing the direction of the second vector and then adding the two vectors. Example: R=A-B can be written as R=A+(-B) where A and B are the vectors and R is the resultant vector.