Question

if magnitude of sum of two unit vestors is root 2
then find the magnitude of
substraction of
these
unit vectors.​

Answer 1

Answer:

If [math]\hat a[/math] and [math]\hat b[/math] are two unit vectors, then the magnitude of their sum would be [math]\sqrt {\|\hat a|^2 + 2\|\hat a|\|\hat b|\cos \theta + \|\hat b|^2}[/math], where [math]\theta[/math] is the angle between the two vectors.

[math]= \sqrt {1 + 2\cos \theta + 1} = \sqrt {2 + 2\cos \theta}[/math]

It is given that the magnitude is [math]\sqrt {2}.[/math]

[math]\Rightarrow \qquad \sqrt {2 + 2\cos \theta} = \sqrt {2} \qquad \Rightarrow \qquad 2 + 2\cos \theta = 2[/math]

[math]\Rightarrow \qquad \cos \theta = 0 \qquad \Rightarrow \qquad \theta = 90^o.[/math]

When we subtract one of these vectors from the other, the angle between one vector and the negative of the other vector, [math]\alpha,[/math] is the supplementary angle of the original angle, [math]\theta.[/math]

[math]\Rightarrow \qquad \alpha = 180^o - \theta = 180^o - 90^o = 90^0.[/math]

So, the magnitude of the resultant vector on subtraction of one of these vectors from the other is given by [math]\sqrt {\|\hat a|^2 + 2\|\hat a|\|\hat b|\cos \alpha + \|\hat b|^2}[/math]

[math]= \sqrt {1 + 2\cos 90^o + 1} = \sqrt {2}[/math]