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Vectors and the Geometry of Three Dimensions

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

7short 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 vector equation of a line?
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A line is fixed by one point on it (position vector a\mathbf{a}) and a direction d\mathbf{d}:
What are the angle between two lines?
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The angle between lines depends only on their directions d1\mathbf{d}_1 and d2\mathbf{d}_2:
What are classifying two lines?
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In three dimensions, two lines are exactly one of:
What is cartesian form with a zero direction component?
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If a direction component is zero, that coordinate is constant; write it as a separate equation rather than dividing by zero.
What is q1?
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Write the vector equation of the line through (0,1,2)(0, 1, 2) with direction (10−1)\begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}. [1 mark]
What is q2?
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Two lines have parallel direction vectors but no common point. What is their relationship? [1 mark]
What is q3?
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How do you confirm two non-parallel lines are skew? [2 marks]

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