Correct Answer - Option 3 : √2, √2
CONCEPT:
The scalar product of two vectors \(\vec a \ and \ \vec b \)is given by \(\vec a \cdot \;\vec b = \left| {\vec a} \right| \times \left| {\vec b} \right|\cos θ \)
CALCULATION:
Given: Vector \(\vec r=a \hat i+b \hat j\) is equally inclined to both x and y axes and the magnitude of the vector is 2 units.
i.e \(|\vec r| = 2\)
⇒ \(\sqrt {a^2 + b^2} = 2\)
⇒ a2 + b2 = 4 --------(1)
∵ The vector \(\vec r=a \hat i+b \hat j\) is equally inclined to both x and y axes
Let θ be the angle between the vector \(\vec r=a \hat i+b \hat j\) and both the x and y axes.
⇒ \(cos \ \theta = \frac{a}{2} = \frac{b}{2}\)
⇒ a = b
So, by substituting a = b in equation (1), we get
⇒ 2b2 = 4 ⇒ b = √2
So, a = b = √2
Hence, correct option is 3