Question

Write two different vectors having the same magnitude.

Answer 1

Hint:
Here we will assume two vectors such that they have the same magnitude. It is not necessary that two vectors having the same magnitude will have the same direction. If two vectors have the same magnitude and same direction, then those vectors are called equal vectors.

Complete step by step solution:
Here we have to write vectors which have the same magnitude.
Let the first vector be \[\overrightarrow a = \widehat i + 2\widehat j + 3\widehat k\] and the second vector be \[\overrightarrow b = 2\widehat i + \widehat j + 3\widehat k\].
We know magnitude of any vector \[\overrightarrow c = x\widehat i + y\widehat j + z\widehat k\] is represented by \[\left| {\overrightarrow c } \right|\] and it is equal to \[\sqrt {{x^2} + {y^2} + {z^2}} \].
Therefore, magnitude of vector \[\overrightarrow a \] can be represented as \[\left| {\overrightarrow a } \right|\] .
Thus,
\[ \Rightarrow \left| {\overrightarrow a } \right| = \sqrt {{1^2} + {2^2} + {3^2}} \]
On further simplifying the terms, we get
\[ \Rightarrow \left| {\overrightarrow a } \right| = \sqrt {1 + 4 + 9} \]
On adding the terms, we get
\[ \Rightarrow \left| {\overrightarrow a } \right| = \sqrt {14} \]
Thus, the magnitude of the vector \[\overrightarrow a \] is \[\sqrt {14} \] .
Now, we will find the magnitude of the vector \[\overrightarrow b = 2\widehat i + \widehat j + 3\widehat k\]
Therefore, magnitude of vector \[\overrightarrow b \] can be represented as \[\left| {\overrightarrow b } \right|\] .
Thus,
\[ \Rightarrow \left| {\overrightarrow b } \right| = \sqrt {{2^2} + {1^2} + {3^2}} \]
On further simplifying the terms, we get
\[ \Rightarrow \left| {\overrightarrow b } \right| = \sqrt {4 + 1 + 9} \]
Adding the terms, we get
\[ \Rightarrow \left| {\overrightarrow b } \right| = \sqrt {14} \]
Thus, the magnitude of the vector \[\overrightarrow b \] is \[\sqrt {14} \] .
Hence, the magnitude of vector \[\overrightarrow a = \widehat i + 2\widehat j + 3\widehat k\] and vector \[\overrightarrow b = 2\widehat i + \widehat j + 3\widehat k\] are equal.

Thus, the required two vectors having same magnitude are \[\overrightarrow a = \widehat i + 2\widehat j + 3\widehat k\] and \[\overrightarrow b = 2\widehat i + \widehat j + 3\widehat k\].

Note:
If two vectors have the same magnitude, they may or may not be equal because for two vectors to be equal, they must have the same direction as well. We can say that equal vectors are vectors which have the same magnitude as well as the same direction. Here in this question also, these two vectors have the same magnitude but they do not have the same direction.
Some important properties of vectors are:-
1) To add the two vectors, we always consider both their magnitudes and directions.
2) Negative vectors are those whose direction is just opposite to the reference positive direction.