Topic 4: Vectors in Three Dimensions

3 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How does the dot product measure the angle between two vectors and detect perpendicularity?

Represent vectors in three dimensions, compute magnitudes and unit vectors, and use the dot product to find angles and test for perpendicularity.

Three-dimensional vectors in component form, magnitude and unit vectors, the dot product in both its component and geometric forms, and using it to find angles, projections and perpendicularity.

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How do we find a vector perpendicular to two others, and what does its length tell us?

Compute the cross product of two vectors, interpret its direction and magnitude, and use it to find areas and normals.

The cross product in three dimensions, why it is perpendicular to both inputs, the right-hand rule, the area-of-parallelogram interpretation of its magnitude, and using it to find normal vectors.

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How do vector and cartesian equations describe lines and planes, and how do we find where they meet?

Write vector, parametric and cartesian equations of lines and planes in three dimensions and find intersections and angles between them.

Vector, parametric and cartesian forms for lines and planes in three dimensions, the role of direction and normal vectors, and finding intersections and angles between lines and planes.

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