Vectors

WFM03Topic 3.53 questions

Vector product, triple scalar product, and equations of lines and planes.

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Question 16 marks
The points $A(1, 0, 2)$ , $B(2, 1, 0)$ and $C(0, 3, 1)$ lie in the plane $\Pi$ .

(a) Find a Cartesian equation of $\Pi$ .

(b) Find the exact perpendicular distance from the point $P(3, 3, 3)$ to $\Pi$ .
Mark scheme
- M1 $\overrightarrow{AB} = (1, 1, -2)$ and $\overset{}{A}$ (1)
Question 23 marks
Find the volume of the tetrahedron with vertices $O(0, 0, 0)$ , $A(1, 0, 2)$ , $B(2, 1, 0)$ and .

C

\to

=

(

-

1

,

3

,

-

1

)

\overrightarrow{AC} = (-1, 3, -1)

A C = (- 1, 3, - 1)

M1 Normal

\mathbf{n} = \overrightarrow{AB} \times \overrightarrow{AC}

(1)

\mathbf{n} = (5, 3, 4)

(1)

5x + 3y + 4z = 13

(1)

M1 Distance

= \dfrac{|5(3) + 3(3) + 4(3) - 13|}{\sqrt{25 + 9 + 16}}

(1)

= \dfrac{23}{\sqrt{50}} = \dfrac{23\sqrt{2}}{10}

(1)

(

)

C(0, 3, 1)

Mark scheme