Further Mathematics 9231: Vectors Topical Past Paper Questions

(a) Find the set of values of k for which the system of equations $$ 3x+k y+7z=1, $$ $$ 8x+\ y+2z=2, $$ $$ 8x+2y+4z=3, $$ has a unique solution and interpret this situation geometrically. [4] (b) Given that $ k = \frac{7}{2} $, show that the system of equations in part 1(a) is in…

The points $A, B, C$ have position vectors $$ 3\mathbf{i}+5\mathbf{j}+5\mathbf{k},\quad2\mathbf{i}+2\mathbf{j}+2\mathbf{k}, $$ respectively, relative to the origin O. (a) Find the equation of the plane $ABC$, giving your answer in the form $ax + by + cz = d$. [5] (b) The point $D…

The lines $ l_1 $ and $ l_2 $ have equations $ \mathbf{r} = \mathbf{i} + 3\mathbf{j} - 2\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} + \mathbf{k}) $ and $ \mathbf{r} = \mathbf{i} - 2\mathbf{j} + 9\mathbf{k} + \mu(\mathbf{i} - 4\mathbf{j} + 2\mathbf{k}) $ respectively. The plane…