| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2008 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Line intersection verification |
| Difficulty | Standard +0.3 This is a standard two-part question on 3D vector lines requiring routine techniques: dot product of direction vectors to show perpendicularity, then solving simultaneous equations to verify no intersection point exists. Both parts follow textbook methods with straightforward algebra, making it slightly easier than average. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use \(3\mathbf{i} - 4\mathbf{j} + 2\mathbf{k}\) and \(2\mathbf{i} - \mathbf{j} - 5\mathbf{k}\) only | M1 | Use correct method for scalar prod of any 2 vectors |
| Obtain \(6 + 4 - 10\), state \(= 0\) & deduce perp | AG, A1 | 3 |
| (ii) Produce 3 equations in \(s\) and \(t\) | *M1 | Of the type \(5 + 3s = 2 + 2t\), \(-2 - 4s = -2 - t\) and \(-2 + 2s = 7 - 5t\) Or Eliminate \(s\) (or \(t\)) from 2 pairs |
| Solve 2 of the equations for \(s\) and \(t\) Obtain \((s,t) = \left(\frac{3}{5}, \frac{12}{5}\right)\) or \(\left(\frac{9}{11}, \frac{33}{22}\right)\) or \(\left(\frac{19}{19}, \frac{33}{19}\right)\) | A1 | dep*M1 |
| Substitute their values in 3rd equation State/show inconsistency & state non-parallel \(\therefore\) skew | dep*M1 | State/show inconsistency & state non-parallel \(\therefore\) skew |
(i) Use $3\mathbf{i} - 4\mathbf{j} + 2\mathbf{k}$ and $2\mathbf{i} - \mathbf{j} - 5\mathbf{k}$ only | M1 | Use correct method for scalar prod of any 2 vectors |
Obtain $6 + 4 - 10$, state $= 0$ & deduce perp | AG, A1 | 3 |
(ii) Produce 3 equations in $s$ and $t$ | *M1 | Of the type $5 + 3s = 2 + 2t$, $-2 - 4s = -2 - t$ and $-2 + 2s = 7 - 5t$ Or Eliminate $s$ (or $t$) from 2 pairs |
Solve 2 of the equations for $s$ and $t$ Obtain $(s,t) = \left(\frac{3}{5}, \frac{12}{5}\right)$ or $\left(\frac{9}{11}, \frac{33}{22}\right)$ or $\left(\frac{19}{19}, \frac{33}{19}\right)$ | A1 | dep*M1 | $(5t=12, 11t=18, 19t=33)$ or $(5s=3, 22s=18, 19s=19)$ A1, A1 |
Substitute their values in 3rd equation State/show inconsistency & state non-parallel $\therefore$ skew | dep*M1 | State/show inconsistency & state non-parallel $\therefore$ skew | A1 | 5 | WWW |5 The vector equations of two lines are
$$\mathbf { r } = ( 5 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } ) + s ( 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = ( 2 \mathbf { i } - 2 \mathbf { j } + 7 \mathbf { k } ) + t ( 2 \mathbf { i } - \mathbf { j } - 5 \mathbf { k } ) .$$
Prove that the two lines are\\
(i) perpendicular,\\
(ii) skew.
\hfill \mbox{\textit{OCR C4 2008 Q5 [8]}}