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Specialist2023 Exam 1 · Q9Tech-free6 marks

Plane through three points, normal vector and parallelogram area

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A plane contains the points $A(1, 3, -2)$ , $C(a, -1, 5)$ and $C(a, -1, 5)$ , where is a real constant. The plane has a -axis intercept of 2 at the point .

a.

Write down the coordinates of point $D$ .

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b.

Show that $\overrightarrow{AB}$ and $\overrightarrow{AD}$ are $-2\tvec{i} - 5\tvec{j} + 6\tvec{k}$ and $-\tvec{i} - \tvec{j} + 2\tvec{k}$ , respectively.

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c.

Hence find the equation of the plane in Cartesian form.

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d.

Find $a$ .

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e.

$\overline{AB}$ and $\overline{AD}$ are adjacent sides of a parallelogram. Find the area of this parallelogram.

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(

a

,

−

1

,

5

)

a

y

D