Can you add three unit vectors to get a unit vector? Does your answer change if two unit vectors are along the coordinate axes?

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Solution

Yes we can add three unit vectors to get a unit vector.
No, the answer does not change if two unit vectors are along the coordinate axes. Assume three unit vectors $\hat{i,} - \hat{i} and \hat{j}$ along the positive x-axis, negative x-axis and positive y-axis, respectively. Consider the figure given below:

The magnitudes of the three unit vectors ( $\hat{i,} - \hat{i} and \hat{j}$ ) are the same, but their directions are different.
So, the resultant of $\hat{i} and - \hat{i}$ is a zero vector.
Now, $\hat{j} + \vec{0} = \hat{j}$ (Using the property of zero vector)
∴ The resultant of three unit vectors ( $\hat{i,} - \hat{i} and \hat{j}$ ) is a unit vector ( $\hat{j}$ ).

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