Edexcel IAL P4 Vectors Questions
Vectors in 2D and 3D; magnitude, unit vectors, scalar product and angles; vector equation of a line; intersecting, parallel and skew lines.
- P4 · Pure Mathematics 4
- Calculator allowed
- 57 practice questions
What the questions test
The 57 P4 vectors practice questions cover:
Worked example 1
Line $L_1$ has vector equation $\mathbf{r_1} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}$ and line $L_2$ has equation $\mathbf{r_2} = \begin{pmatrix} 5 \\ -3 \\ 0 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix}$.
- Show that $L_1$ and $L_2$ intersect.
- Find the exact coordinates of the point of intersection.
Mark scheme
- M1 for equating the parametric equations: $1 + 2\lambda = 5 - \mu$, $\lambda = -3 + 2\mu$, and $1 + \lambda = \mu$.
M1 for taking two equations to solve for the parameters, e.g., $\lambda = \mu - 1$ into $2\lambda + \mu = 4$.
M1 for $2(\mu - 1) + \mu = 4 \implies 3\mu - 2 = 4 \implies 3\mu = 6$.
A1 for finding $\mu = 2$ and $\lambda = 1$.
B1 for verifying with the unused equation: $\lambda - 2\mu = 1 - 4 = -3$, which matches the second component equation, showing the lines intersect. - M1 for substituting $\lambda = 1$ into $L_1$ (or $\mu = 2$ into $L_2$).
A1 for the intersection point $(3, 1, 2)$.
Worked example 2
Consider the following two lines in 3D space: \[L_1: r = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix} \quad \text{and} \quad L_2: r = \begin{pmatrix} 4 \\ 2 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix}\] Show algebraically that these two lines are skew (they are neither parallel nor do they intersect).
Mark scheme
Set the components equal:
\(1 + 2\lambda = 4 - \mu \implies 2\lambda + \mu = 3\) (Eq 1). M1
\(-1 + \lambda = 2 + 2\mu \implies \lambda - 2\mu = 3\) (Eq 2). M1
\(2 - 3\lambda = -1 + \mu \implies -3\lambda - \mu = -3\) (Eq 3). M1
Solve Eq 1 and Eq 2: \(2(3 + 2\mu) + \mu = 3 \implies 6 + 5\mu = 3 \implies \mu = -0.6\). M1
Substitute \(\mu = -0.6 \implies \lambda = 1.8\). A1
Check in Eq 3: \(-3(1.8) - (-0.6) = -5.4 + 0.6 = -4.8\). M1
However, the RHS of Eq 3 is \(-3\). Since \(-4.8 \neq -3\), there is no consistent solution. B1
Therefore, the lines do not intersect. Since their direction vectors are not scalar multiples, they are skew. A1
FAQ
What vectors is in Edexcel IAL P4?
Vectors in 2D and 3D; magnitude, unit vectors, scalar product and angles; vector equation of a line; intersecting, parallel and skew lines.
Is P4 a calculator paper?
Yes, a calculator is allowed in P4, but method marks are only given for working you write down.
How are P4 vectors questions marked?
With M marks for a correct method, A marks for accurate answers that depend on the M mark before them, and B marks for independent results, as in the worked examples on this page.