Question

What is the direction of the vector A plus vector B ?
\[\vec A = 3\hat i + 2\hat j\] and \[\vec B = - \hat 1 - 2\hat j\]

Answer 1

Hint: We are asked to find the resultant of two given vectors. We can find the resultant of the two vectors by merely adding them and we get a value. After we do this process, we can see the direction of the vector. The direction of the vector is represented using unit vectors. Then we see what unit vector is obtained and we can compare the unit vector with the axes and finally get to our required solution.

Formulas used:
To compare the axes with the unit vectors we use unit vectors for each direction and it is as follows:
The direction of the x-axis is represented by \[x - axis \Rightarrow \hat i\].
The direction of the y-axis is represented by \[y - axis \Rightarrow \hat j\].
The direction of the z-axis is represented by \[z - axis \Rightarrow \hat k\].

Complete step by step answer:
We can start by noting down the data given in the question:
\[\vec A = 3\hat i + 2\hat j\]
\[\Rightarrow \vec B = - \hat 1 - 2\hat j\]
Let us add the two and see what the resultant will be
\[\vec A + \vec B = 3\hat i + 2\hat j + \left( { - 1\hat i - 2\hat j} \right) \\
\therefore vec A + \vec B = 2\hat i\]
We compare it with the direction of the $x$ axis and now we get to know that the direction is in the $x$ direction.

Therefore, the resultant of the two given vectors A and B is in the $x$ direction.

Note: Vectors are the mathematical representation of a quantity that has both direction and magnitude. Vectors play a very important role in physics as they are used to represent a wide number of quantities like position (the position vector), displacement, velocity acceleration etc. Vectors help us understand physical quantities better. Another application of vectors comes in the branch of engineering mechanics.