Question

If the sum of two unit vectors is also a vector of unit magnitude, the magnitude of the difference of the two-unit vectors is:
A) $1unit$
B) $2units$
C) $\sqrt 3 units$
D) $Zero$

Answer 1

Hint: We know a vector having unit magnitude and a direction, which is referred to as unit vectors. We know vectors can be added using the triangle law of parallelogram law. We can use the formula of addition to obtaining the value of the dot products of the two vectors, and this value is then used to find the difference between the two vectors.

Formula used:
Addition of vectors:
$|(\vec a + \vec b){|^2} = |\vec a| + \vec b| + 2.\vec a.\vec b$
Subtraction of vectors:
$|(\vec a - \vec b){|^2} = |\vec a| + \vec b| - 2.\vec a.\vec b$
Where:
$\vec a$ and $\vec b$ are unit vectors.

Complete step by step solution:
When two or more vectors are added, the sum of the vectors is referred to as the resultant vector.
In the question, we are given that the resultant vector of the addition of two vectors is also a vector; thus, we use the formula:
$\Rightarrow |(\vec a + \vec b){|^2} = |\vec a| + \vec b| + 2.\vec a.\vec b$
Let us consider the resultant vector as $\vec c$, and as we know,
$\Rightarrow |\vec c| = |\vec a + \vec b|^2 = 1$ and
since $\vec a$ and $\vec b$ are unit vectors, we write $|a| = |b| = 1$
Therefore, when we put the values in the equation, we get:
$\Rightarrow {1^2} = 1 + 1 + 2.\vec a.\vec b$
Thus, on solving this equation, we get:
$\Rightarrow \vec a.\vec b = - \dfrac{1}{2}$
Now, as per the question, we need to find the difference between the two unit vectors:
$\Rightarrow |(\vec a - \vec b){|^2} = 1 + 1 - 2.\vec a.\vec b$
Now, putting the value of $\vec a.\vec b$ as obtained above, we obtain:
$\Rightarrow |(\vec a - \vec b){|^2} = 1 + 1 + 2 \times \dfrac{1}{2}$
Hence, on solving, we get:
$\Rightarrow |(\vec a - \vec b){|^2} = 3$
Therefore,
$\Rightarrow |(\vec a - \vec b){|^{}} = \sqrt 3$

Thus, option (C) is correct.

Note: The Triangle law of vector addition states that if two vectors represent two sides of a triangle, in both order and magnitude, then the third side will represent the magnitude and direction of the resultant vector. The other law is the Parallelogram law, which states if two vectors represent two adjacent sides of a parallelogram, the diagonal of the parallelogram represents the resultant vector.