**Topic: Vector and Parametric Equations****Objective:**Find the vector and parametric equations for the line passing through the point \( P(4, -2, -4) \) and parallel to the vector \( 4\mathbf{i} - \mathbf{j} + 4\mathbf{k} \).---**Vector Form Equation:**\[\mathbf{r} = \langle 4, -2, -4 \rangle + t \langle 4, -1, 4 \rangle\]- **Initial Point**: \(\langle 4, -2, -4 \rangle\)- **Direction Vector**: \(\langle 4, -1, 4 \rangle\) ---**Parametric Form Equations:**Using parameter \( t \):- \( x = x(t) = 4 + 4t \)- \( y = y(t) = -2 - t \)- \( z = z(t) = -4 + 4t \)These equations represent the line in terms of the parameter \( t \), passing through the point \( P \) when \( t = 0 \).

Find the vector and parametric equations for the linethrough the point P4,-2,4 and parallel to the vector4i^-j^+4k^.To find:1The vector form r→2The parametric formparameter t and passing through P when t=0.The equation of line passing through the point Px0,y0,z0 andparallel to the vector b→ isr→=a→+tb→,where t is a parameter.Here,A vector is passing through the point P4,-2,4 isa→=4i^-2j^+4k^.and a vector r→ is parallel to the vector 4i^-j^+4k^. Then,b→=4i^-j^+4k^.Hence,the vector equation of line r→ becomesr→=4i^-2j^+4k^+t4i^-j^+4k^.bThe parametric equation of line isr→=4i^-2j^+4k^+t4i^-j^+4k^.Hence,xt=4+4t,yt=-2-t and zt=4+4t⇒xt,yt,zt=4+4t,-2-t,4+4t.Now,the parametric equation of line at t=0 isxt,yt,zt=4,-2,4.