Let α =4 i +3 j +5 k and β = i +2 j -4 k. Let β_1 be parallel to α and β_2 be perpendicular to α. If β = β_1+ β_2, then the value of 5 β_2 ·( i + j +…

Let $\displaystyle \vec{\alpha}=4 \hat{i}+3 \hat{j}+5 \hat{k}$ and $\displaystyle \vec{\beta}=\hat{i}+2 \hat{j}-4 \hat{k}$. Let $\displaystyle \vec{\beta}_1$ be parallel to $\displaystyle \vec{\alpha}$ and $\displaystyle \vec{\beta}_2$ be perpendicular to $\displaystyle \vec{\alpha}$. If $\displaystyle \vec{\beta}=\vec{\beta}_1+\vec{\beta}_2$, then the value of $\displaystyle 5 \vec{\beta}_2 \cdot(\hat{i}+\hat{j}+\hat{k})$ is