Find the scalar magnitude of the projection of $2 i-3 j+4 k$ on $i-j$ Express as a Cartesian vector: $\vec{a}=(2,-3,4) \quad$ and $\quad \vec{b}=(1,-1,0)$

Step 1 ：Find the dot product of the two vectors: \(\vec{a} \cdot \vec{b} = (2)(1) + (-3)(-1) + (4)(0) = 5\)

Step 2 ：Find the magnitude of vector \(\vec{b}\): \(||\vec{b}|| = \sqrt{(1)^2 + (-1)^2 + (0)^2} = \sqrt{2}\)

Step 3 ：Calculate the scalar magnitude of the projection: \(\frac{\vec{a} \cdot \vec{b}}{||\vec{b}||} = \frac{5}{\sqrt{2}} = \frac{5\sqrt{2}}{2}\)

Step 4 ：\(\boxed{\text{Final Answer: }\frac{5\sqrt{2}}{2}}\)