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Vectors and the Geometry of Three Dimensions

Quick questions on Intersections and distances explained: H2 Further Mathematics

9short 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 are intersection of two planes?
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Two non-parallel planes meet in a line. The line's direction is perpendicular to both normals, so it is n1×n2\mathbf{n}_1\times\mathbf{n}_2. To find a point on the line, set one coordinate (say z=0z = 0) and solve the two plane equations simultaneously for the other two.
What is perpendicular distance from a point to a plane?
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For a point (x0,y0,z0)(x_0, y_0, z_0) and plane ax+by+cz=dax + by + cz = d,
What is perpendicular distance from a point to a line?
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For a point PP and a line through AA with direction d\mathbf{d}, the distance is
What are shortest distance between two skew lines?
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For skew lines through A1,A2A_1, A_2 with directions d1,d2\mathbf{d}_1, \mathbf{d}_2, the common perpendicular has direction d1×d2\mathbf{d}_1\times\mathbf{d}_2, and the shortest distance is the projection of A1A2→\overrightarrow{A_1 A_2} onto this unit perpendicular:
What is wrong sign handling in the point-to-plane formula?
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Use ax0+by0+cz0−dax_0 + by_0 + cz_0 - d inside the modulus; forgetting the −d-d or the absolute value gives a wrong distance.
What is parallel lines in the skew formula?
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If d1×d2=0\mathbf{d}_1\times\mathbf{d}_2 = \mathbf{0} the lines are parallel and the skew formula breaks down; use the point-to-line distance instead.
What is q1?
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How do you find where a line meets a plane? [2 marks]
What is q2?
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Write the perpendicular distance from (x0,y0,z0)(x_0, y_0, z_0) to the plane ax+by+cz=dax + by + cz = d. [1 mark]
What is q3?
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What direction is the common perpendicular of two skew lines with directions d1\mathbf{d}_1 and d2\mathbf{d}_2? [1 mark]

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