Question

Find the sum of three vectors?

Answer 1

Hint: First, let us assume those three vectors in terms of i, j, and k. To solve the question, we need to add all the i components of the vectors separately then j components, and finally the k components of all the three vectors.

Complete step by step answer:
Here, let us assume the three vectors.
The first vector is $a = {x_1}i + {y_1}j + {z_1}k$, the second vector is $b = {x_2}i + {y_2}j + {z_2}k$ , and the third vector is $c = {x_3}i + {y_3}j + {z_3}k$.
Let us apply the addition of two vectors a, and b. For that, we need to add all the i components of the vectors separately then j components, and finally the k components of all the three vectors.
Therefore,
$ \Rightarrow a + b = \left( {{x_1}i + {y_1}j + {z_1}k} \right) + \left( {{x_2}i + {y_2}j + {z_2}k} \right)$
That is equal to,
$ \Rightarrow a + b = \left( {{x_1} + {x_2}} \right)i + \left( {{y_1} + {y_2}} \right)j + \left( {{z_1} + {z_2}} \right)k$
Now, let us add vector c with the above expression.
$ \Rightarrow a + b + c = \left[ {\left( {{x_1} + {x_2}} \right)i + \left( {{y_1} + {y_2}} \right)j + \left( {{z_1} + {z_2}} \right)k} \right] + \left( {{x_3}i + {y_3}j + {z_3}k} \right)$
That is equal to,
$ \Rightarrow a + b + c = \left( {{x_1} + {x_2} + {x_3}} \right)i + \left( {{y_1} + {y_2} + {y_3}} \right)j + \left( {{z_1} + {z_2} + {z_3}} \right)k$
Hence, the sum of three vectors $a = {x_1}i + {y_1}j + {z_1}k$, $b = {x_2}i + {y_2}j + {z_2}k$, and $c = {x_3}i + {y_3}j + {z_3}k$ is $a + b + c = \left( {{x_1} + {x_2} + {x_3}} \right)i + \left( {{y_1} + {y_2} + {y_3}} \right)j + \left( {{z_1} + {z_2} + {z_3}} \right)k$ .

Note: The vector addition is different from the normal addition because vectors have direction. Here, i, j, and k represent x, y, and z directions respectively. So, while adding or subtracting the vectors we have to add or subtract the value of the same direction only. We should also be careful about the sign. Therefore, we must take care of the quadrant in which the vector lies.
We can directly add the three vectors of the given question.
The first vector is $a = {x_1}i + {y_1}j + {z_1}k$, the second vector is $b = {x_2}i + {y_2}j + {z_2}k$ , and the third vector is $c = {x_3}i + {y_3}j + {z_3}k$.
Let us apply the addition of two vectors a, and b. For that, we need to add all the i components of the vectors separately then j components, and finally the k components of all the three vectors.
Therefore,
$ \Rightarrow a + b + c = \left( {{x_1}i + {y_1}j + {z_1}k} \right) + \left( {{x_2}i + {y_2}j + {z_2}k} \right) + \left( {{x_3}i + {y_3}j + {z_3}k} \right)$
That is equal to,
$ \Rightarrow a + b + c = \left( {{x_1} + {x_2} + {x_3}} \right)i + \left( {{y_1} + {y_2} + {y_3}} \right)j + \left( {{z_1} + {z_2} + {z_3}} \right)k$