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Topic 3: Vectors in two and three dimensions

Vectors in three dimensions

3 hours

• Use Cartesian coordinates for three-dimensional space, including plotting points. 1 interactive
• Use ordered triple notation (x, y, z) and column vector notation to represent a position vector in three dimensions.
• Calculate the magnitude of a vector.
• Calculate and use a unit vector, â, in three-dimensional space. 1 interactive
• Define and use unit vectors and the perpendicular unit vectors î, ĵ and k̂.
• Express a vector in Cartesian (component) form using the unit vectors î, ĵ and k̂.
• Define and use the altitude angle φ. 1 interactive

Algebra of vectors in three dimensions

3 hours

• Examine and use addition and subtraction of vectors in Cartesian form.
• Use multiplication by a scalar of a vector in Cartesian form.
• Determine a vector between two points.
• Use a vector representing a section of a line segment, including the midpoint of a line segment.
• Use the scalar (dot) product.
• Examine properties of parallel and perpendicular vectors and determine if two vectors are parallel or perpendicular.
• Use scalar and vector projections of vectors.
• Apply the scalar product to vectors expressed in Cartesian form.
• Model and solve problems that involve displacement, force, velocity and relative velocity using the above concepts.
• Use vectors to prove geometric results in two dimensions (other than those listed in Unit 2 Topic 3) and in three dimensions.

Vector and Cartesian equations

7 hours

• Understand and use equations of spheres. 1 interactive
• Use vector equations of curves in two or three dimensions involving a parameter, and determine a ‘corresponding’ Cartesian equation in the two-dimensional case.
• Determine vector, parametric and Cartesian equations of straight lines and straight-line segments given the position of two points, or equivalent information, in both two and three dimensions. 1 interactive
• Define and use the vector (cross) product to determine a vector normal to a given plane, with and without technology. 1 interactive
• Use vector methods in applications, including areas of shapes and determining vector and Cartesian equations of a plane and of regions in a plane.