What will be the projection of vector $$\vec{A} = \hat{i} + \hat{j} + \hat{k}$$ on vector $$\vec{B} = \hat{i} + \hat{j}$$?

The vector projection of $$\vec{A}$$ onto $$\vec{B}$$ is given by the formula $$\text{proj}_{\vec{B}}\vec{A} = \dfrac{\vec{A} \cdot \vec{B}}{|\vec{B}|^2}\,\vec{B}$$.

First, compute the dot product $$\vec{A} \cdot \vec{B} = (1)(1) + (1)(1) + (1)(0) = 2$$.

Next, find the magnitude squared of $$\vec{B}$$: $$|\vec{B}|^2 = 1^2 + 1^2 = 2$$.

Therefore the projection is $$\dfrac{2}{2}(\hat{i} + \hat{j}) = \hat{i} + \hat{j}$$.