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Vectors and the Geometry of Three Dimensions

Quick questions on Planes in three dimensions explained: H2 Further Mathematics

8short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the normal vector defines a plane?
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A plane is fixed by one point on it (position vector a\mathbf{a}) and a normal vector n\mathbf{n} perpendicular to it. A point r\mathbf{r} lies in the plane exactly when r−a\mathbf{r} - \mathbf{a} is perpendicular to n\mathbf{n}, that is (r−a)⋅n=0(\mathbf{r} - \mathbf{a})\cdot\mathbf{n} = 0.
What is the vector (parametric) form?
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A plane can also be written with two direction vectors u\mathbf{u} and v\mathbf{v} lying in it:
What are finding a normal from points?
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Given three points A,B,CA, B, C in the plane, form two directions AB→\overrightarrow{AB} and AC→\overrightarrow{AC} and take their cross product for the normal. Then use any one point to find dd.
What are angles?
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The angle between two planes equals the angle between their normals:
What is wrong dd from the wrong point?
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Compute d=a⋅nd = \mathbf{a}\cdot\mathbf{n} from a point that actually lies in the plane.
What is q1?
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State a normal vector to the plane 3x+2y−z=73x + 2y - z = 7. [1 mark]
What is q2?
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How do you find a normal to a plane through three given points? [2 marks]
What is q3?
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Which trig function relates the line-plane angle to the dot product of direction and normal? [1 mark]

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