Question

\[ 1. r(u,v) = (u + v)i + (3 - v)j + (1 + 4u + 5v)k\]

Short answer

PLANE

Step-by-step solution

Step 1: To Find the vector equation

Remember that

The vector equation of a plane containing the vectors \({b_1}\)and \({b_2}\)and containing a point with position vector a is

\(r(u,v) = a + u{b_1} + v{b_2} \to {\rm{ Equation }}1\)

We can Rewrite the given vector equation as follows:

\(\begin{aligned}{l}r(u,v) &= (u + v)i + (3 - v)j + (1 + 4u + 5v)k\\r(u,v) &= (3j + k) + u(i + 4k) + v(i - j + 5k) \to {\rm{ Equation }}2\end{aligned}\)

Therefore, the given vector equation represents a plane passing through \((0,3,1)\)which contains the vectors \(i + 4k\)and \(i - j + 5k\)

Step 2: Final Answer

PLANE