Pearson Edexcel IAL Mathematics FP3.5.3 Equation of a plane

Practise forming equations of planes in Cartesian, scalar product and parametric forms from points, directions and normal vectors.

Exam points

find a normal vector from two directions, then form a Cartesian plane equation
convert a plane between r·n=p and r=a+sb+tc when points and directions are given
use a plane equation with a given point or line condition to determine an unknown constant

[Maximum number: 3]

The skew lines $l_{1}$ and $l_{2}$ have equations

l_{1}: \mathbf{r}=(\mathbf{i}+2 \mathbf{j}-5 \mathbf{k})+\lambda(5 \mathbf{i}+\mathbf{j})

and

l_{2}: \mathbf{r}=(2 \mathbf{i}-4 \mathbf{j}+4 \mathbf{k})+\mu(8 \mathbf{i}-2 \mathbf{j}+3 \mathbf{k})

where $\lambda$ and $\mu$ are scalar parameters.

(a)

Determine an equation of the plane parallel to $l_{1}$ that contains $l_{2}$

[ 3 ]

(i)

in the form r=a+s b+t c

[ 1 ]

$\mathbf r=(2\mathbf i-4\mathbf j+4\mathbf k) +s(8\mathbf i-2\mathbf j+3\mathbf k) +t(5\mathbf i+\mathbf j)$

B1

(ii)

in the form r.n =p

[ 2 ]

$(2\mathbf i-4\mathbf j+4\mathbf k)\cdot(3\mathbf i-15\mathbf j-18\mathbf k)=-6$

M1

$\mathbf r\cdot(3\mathbf i-15\mathbf j-18\mathbf k)=-6$
or
$\mathbf r\cdot(-\mathbf i+5\mathbf j+6\mathbf k)=2$

A1

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FP3.5.3 - Equation of a plane question 1

Question (a)

Question (i)

Question (ii)