| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Angle between two lines |
| Difficulty | Moderate -0.3 This is a standard three-part vectors question requiring routine techniques: finding a direction vector and line equation, substituting a point to find constants, and using the scalar product formula for angle between lines. All methods are textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10d Vector operations: addition and scalar multiplication4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles |
4. The line $l _ { 1 }$ passes through the points $P$ and $Q$ with position vectors ( $- \mathbf { i } - 8 \mathbf { j } + 3 \mathbf { k }$ ) and ( $2 \mathbf { i } - 9 \mathbf { j } + \mathbf { k }$ ) respectively, relative to a fixed origin.\\
(i) Find a vector equation for $l _ { 1 }$.
The line $l _ { 2 }$ has the equation
$$\mathbf { r } = ( 6 \mathbf { i } + a \mathbf { j } + b \mathbf { k } ) + t ( \mathbf { i } + 4 \mathbf { j } - \mathbf { k } )$$
and also passes through the point $Q$.\\
(ii) Find the values of the constants $a$ and $b$.\\
(iii) Find, in degrees to 1 decimal place, the acute angle between lines $l _ { 1 }$ and $l _ { 2 }$.\\
\hfill \mbox{\textit{OCR C4 Q4 [9]}}