Q.Express the vector $\bar a=5\hat i-2\hat j+5\hat k$ as a sum of two vectors such that one is parallel to the vector $\bar b=3\hat i+\hat k$ and other is perpendicular to $\bar b$.

$\bar a=5\hat i-2\hat j+5\hat k,\ \bar b=3\hat i+\hat k$. The component of $\bar a$ parallel to $\bar b$ is

the vector projection:

$\bar a\cdot\bar b=5(3)+(-2)(0)+5(1)=15+5=20$. $|\bar b|^2=9+0+1=10$.

The perpendicular component is the remainder:

$\bar a_\perp=\bar a-\bar a_\parallel=(5\hat i-2\hat j+5\hat k)-(6\hat i+0\hat j+2\hat k)=-\hat i-2\hat j+3\hat k.$ …

Compute the vector projection of a onto b for the parallel part, subtract it from a to get the perpendicular part, and verif …

Forgetting that b has a zero j-component, and mistakenly including a j-term w …