What is the component of the vector $\vec{A} = 2\hat{i} + 3\hat{j}$ along the direction of the vecto

Question

What is the component of the vector $\vec{A} = 2\hat{i} + 3\hat{j}$ along the direction of the vector $\hat{i} + \hat{j}$?

Professor PC Thomas Classes · Answer

The component of $\vec{A}$ along $\vec{B}$ is given by $\vec{A} \cdot \hat{B}$, where $\hat{B}$ is the unit vector of $\vec{B} = \hat{i} + \hat{j}$. $\hat{B} = \frac{\hat{i} + \hat{j}}{\sqrt{2}}$. Component = $(2\hat{i} + 3\hat{j}) \cdot \left(\frac{\hat{i} + \hat{j}}{\sqrt{2}}\right) = \frac{2 + 3}{\sqrt{2}} = \frac{5}{\sqrt{2}}$.