| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Topic | Vectors 3D & Lines |
| Type | Show lines intersect and find intersection point |
| Difficulty | Standard +0.3 This is a standard two-part vector lines question requiring equating components to find intersection (routine algebraic manipulation) and then using the scalar product formula for angle between lines (direct formula application). Both parts are textbook exercises with no novel insight required, 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 |
**Question 6:**
**(i)** $4\lambda - 3\mu = 24$
$7\lambda - \mu = -9$
$8\lambda - 4\mu = 24$ **M1** (Equate components and form 3 simultaneous equations)
Solve any pair of the simultaneous equations **M1**
$\lambda = -3,\ \mu = -12$ **A1** (Award for $\lambda$ and $\mu$ correct)
$8(-3) - 4(-12) = 24$ so the lines intersect **B1** (Substitute values into the third equation and make explicit conclusion)
$(-15, -10, -33)$ or $\mathbf{r} = -15\mathbf{i} - 10\mathbf{j} - 33\mathbf{k}$ **A1**
**(ii)** $\theta = \cos^{-1}\frac{(4\mathbf{i}+7\mathbf{j}+8\mathbf{k})\bullet(3\mathbf{i}+\mathbf{j}+4\mathbf{k})}{\sqrt{4^2+7^2+8^2}\sqrt{3^2+1^2+4^2}}$ **M1** (Use of correct formula)
$\theta = \cos^{-1}\frac{51}{\sqrt{129}\sqrt{26}}$ **A1** (Correct direction vector values used)
$28.3°$ **A1** (Award also for $0.494$)
**Total: 8 marks**6 Two straight lines have equations
$$\mathbf { r } = - 3 \mathbf { i } + 11 \mathbf { j } - 9 \mathbf { k } + \lambda ( 4 \mathbf { i } + 7 \mathbf { j } + 8 \mathbf { k } )$$
and
$$\mathbf { r } = 21 \mathbf { i } + 2 \mathbf { j } + 15 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )$$
(i) Show that the lines intersect and find the coordinates of their point of intersection.\\
(ii) Find the acute angle between the two lines.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2018 Q6 [8]}}