| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | March |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Vector equation of a line |
| Difficulty | Standard +0.3 This is a straightforward vectors question requiring standard techniques: (i) checking if direction vector is perpendicular to normal vector (simple dot product), and (ii) finding a line in a plane perpendicular to a given line using cross product. Both parts are routine applications of well-practiced methods with no novel insight required, making it slightly easier than average. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms |
| Answer | Marks | Guidance |
|---|---|---|
| (i) EITHER: Substitute for \(r\) in the given equation of \(p\) and expand scalar product | M1 | |
| Obtain equation in \(\lambda\) in any correct form | A1 | |
| Verify this is not satisfied for any value of \(\lambda\) | A1 | |
| OR1: Substitute coordinates of a general point of \(l\) in the Cartesian equation of plane \(p\) | M1 | |
| Obtain equation in \(\lambda\) in any correct form | A1 | |
| Verify this is not satisfied for any value of \(\lambda\) | A1 | |
| OR2: Expand scalar product of the normal to \(p\) and the direction vector of \(l\) | M1 | |
| Verify scalar product is zero | A1 | |
| Verify that one point of \(l\) does not lie in the plane | A1 | |
| OR3: Use correct method to find the perpendicular distance of a general point of \(l\) from \(p\) | M1 | |
| Obtain a correct unsimplified expression in terms of \(\lambda\) | A1 | |
| Show that the perpendicular distance is \(5/\sqrt{6}\), or equivalent, for all \(\lambda\) | A1 | |
| OR4: Use correct method to find the perpendicular distance of a particular point of \(l\) from \(p\) | M1 | |
| Show that the perpendicular distance is \(5/\sqrt{6}\), or equivalent | A1 | |
| Show that the perpendicular distance of a second point is also \(5/\sqrt{6}\), or equivalent | A1 | [3] |
| (ii) EITHER: Calling the unknown direction vector \(a\mathbf{i} + b\mathbf{j} + c\mathbf{k}\) state equation \(2a + b + 3c = 0\) | B1 | |
| State equation \(2a - b - c = 0\) | B1 | |
| Solve for one ratio, e.g. \(a : b\) | M1 | |
| Obtain ratio \(a : b : c = 1 : 4 : -2\), or equivalent | A1 | |
| OR: Attempt to calculate the vector product of the direction vector of \(l\) and the normal vector of the plane \(p\), e.g. \((2\mathbf{i} + \mathbf{j} + 3\mathbf{k}) \times (2\mathbf{i} - \mathbf{j} - \mathbf{k})\) | M2 | |
| Obtain two correct components of the product | A1 | |
| Obtain answer \(2\mathbf{i} + 8\mathbf{j} - 4\mathbf{k}\), or equivalent | A1 | |
| Form line equation with relevant vectors | M1 | |
| Obtain answer \(\mathbf{r} = 5\mathbf{i} + 3\mathbf{j} + \mathbf{k} + \mu(4\mathbf{j} - 2\mathbf{k})\), or equivalent | A1* | [6] |
(i) **EITHER:** Substitute for $r$ in the given equation of $p$ and expand scalar product | M1 |
Obtain equation in $\lambda$ in any correct form | A1 |
Verify this is not satisfied for any value of $\lambda$ | A1 |
**OR1:** Substitute coordinates of a general point of $l$ in the Cartesian equation of plane $p$ | M1 |
Obtain equation in $\lambda$ in any correct form | A1 |
Verify this is not satisfied for any value of $\lambda$ | A1 |
**OR2:** Expand scalar product of the normal to $p$ and the direction vector of $l$ | M1 |
Verify scalar product is zero | A1 |
Verify that one point of $l$ does not lie in the plane | A1 |
**OR3:** Use correct method to find the perpendicular distance of a general point of $l$ from $p$ | M1 |
Obtain a correct unsimplified expression in terms of $\lambda$ | A1 |
Show that the perpendicular distance is $5/\sqrt{6}$, or equivalent, for all $\lambda$ | A1 |
**OR4:** Use correct method to find the perpendicular distance of a particular point of $l$ from $p$ | M1 |
Show that the perpendicular distance is $5/\sqrt{6}$, or equivalent | A1 |
Show that the perpendicular distance of a second point is also $5/\sqrt{6}$, or equivalent | A1 | [3]
(ii) **EITHER:** Calling the unknown direction vector $a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$ state equation $2a + b + 3c = 0$ | B1 |
State equation $2a - b - c = 0$ | B1 |
Solve for one ratio, e.g. $a : b$ | M1 |
Obtain ratio $a : b : c = 1 : 4 : -2$, or equivalent | A1 |
**OR:** Attempt to calculate the vector product of the direction vector of $l$ and the normal vector of the plane $p$, e.g. $(2\mathbf{i} + \mathbf{j} + 3\mathbf{k}) \times (2\mathbf{i} - \mathbf{j} - \mathbf{k})$ | M2 |
Obtain two correct components of the product | A1 |
Obtain answer $2\mathbf{i} + 8\mathbf{j} - 4\mathbf{k}$, or equivalent | A1 |
Form line equation with relevant vectors | M1 |
Obtain answer $\mathbf{r} = 5\mathbf{i} + 3\mathbf{j} + \mathbf{k} + \mu(4\mathbf{j} - 2\mathbf{k})$, or equivalent | A1* | [6]8 The line $l$ has equation $\mathbf { r } = \left( \begin{array} { r } 1 \\ 2 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right)$. The plane $p$ has equation $\mathbf { r } \cdot \left( \begin{array} { r } 2 \\ - 1 \\ - 1 \end{array} \right) = 6$.\\
(i) Show that $l$ is parallel to $p$.\\
(ii) A line $m$ lies in the plane $p$ and is perpendicular to $l$. The line $m$ passes through the point with coordinates (5, 3, 1). Find a vector equation for $m$.
\hfill \mbox{\textit{CAIE P3 2016 Q8 [9]}}