Edexcel IAL Maths FP3.5.2 Vector Problems

[Maximum number: 7]

The plane $\Pi_1$ has vector equation

\mathbf r=\begin{smallmatrix}5\\3\\0\end{smallmatrix}+s\begin{smallmatrix}3\\0\\1\end{smallmatrix}+t\begin{smallmatrix}1\\-2\\2\end{smallmatrix}

where s and t are scalar parameters.

(a)

The plane $\Pi_2$ has vector equation $\mathbf r\cdot\begin{smallmatrix}5\\-2\\3\end{smallmatrix}=1$ .

Determine a vector equation for the line of intersection of $\Pi_1$ and $\Pi_2$ .

Give your answer in the form $\mathbf r=\mathbf a+\lambda\mathbf b$ , where $\mathbf a$ and $\mathbf b$ are constant vectors and $\lambda$ is a scalar parameter.

[ 4 ]

From part (a), $\Pi_1:2x-5y-6z=-5$ , and $\Pi_2:5x-2y+3z=1$ .

Taking y=0,

2x-6z=-5,\quad 5x+3z=1

x=-\frac14,\quad z=\frac34

A point on the line is $(-\frac14,0,\frac34)$ . A direction vector is

\begin{smallmatrix}2\\-5\\-6\end{smallmatrix}\times\begin{smallmatrix}5\\-2\\3\end{smallmatrix}=\begin{smallmatrix}-27\\-36\\21\end{smallmatrix}

so an equivalent direction vector is $\begin{smallmatrix}9\\12\\-7\end{smallmatrix}$ . Therefore

\mathbf r=\begin{smallmatrix}-\frac14\\0\\\frac34\end{smallmatrix}+\lambda\begin{smallmatrix}9\\12\\-7\end{smallmatrix}

M1/dM1: finds a point on the line of intersection.
ddM1: forms a direction vector from the plane normals.
A1: any correct vector equation.

(b)

The plane $\Pi_3$ has Cartesian equation 4x-3y-z=0.

Use the answer to part (b) to determine the coordinates of the point of intersection of $\Pi_1$ , $\Pi_2$ and $\Pi_3$ .

[ 3 ]

Using the line from part (b),

\mathbf r=\begin{smallmatrix}-\frac14\\0\\\frac34\end{smallmatrix}+\lambda\begin{smallmatrix}9\\12\\-7\end{smallmatrix}

x=-\frac14+9\lambda,\quad y=12\lambda,\quad z=\frac34-7\lambda

Substitute into 4x-3y-z=0:

4\left(-\frac14+9\lambda\right)-3(12\lambda)-\left(\frac34-7\lambda\right)=0

7\lambda-\frac74=0\Rightarrow \lambda=\frac14

(x,y,z)=(2,3,-1)

M1: substitutes the parametric line from part (b) into Pi_3 and solves for the parameter.
dM1: substitutes the parameter into the line.
A1: correct point only.

Edexcel A-Level Mathematics A2 Fp3 5 2 Use of Vectors in Problems Questions

Exam points

Edexcel A-Level Mathematics A2 Fp3 5 2 Use of Vectors in Problems Questions question 1

Question (a)

Question (b)