TEACHING POINTS Begin by showing some 3D coordinates on axes. (Graph drawing packages are very useful here, x, y, z especially if you can turn the grid to view from different positions.) Consider a cuboid (2 by 3 by 4), with one corner at the origin. Ask the class to write down the coordinates of all the vertices. Remind students of 2D work and extend to 3D column vectors, orthogonal unit vectors i, j, k and position vectors. Write all the vectors from the corner of the cuboid as position vectors (e.g. OA = a, etc.) ‘origin’ Calculate the magnitude of these vectors as for example. Extend this idea to calculating the distance between two points (x1, and (x2, using d y1, z1) y2, z2) 2 = (x1 + (y1 + (z1 d – x2)2 – y2)2 – z2)2 Extend the ideas of vector addition and subtraction to 3D: = b a. – Cover the triangle and parallelogram laws of addition, as well as demonstrating parallel vectors. Show how to find a unit vector in the direction of a, and make sure students are familiar with the notationa (extended to 3D). Use vectors to solve problems in pure mathematics and discuss the 3D geometrical interpretations of solutions. OPPORTUNITIES FOR REASONING/PROBLEM SOLVING Link examples to mechanics (kinematics and forces). For example, consider questions such as: The velocity of an object is given by vector = 3ti + + 4k v t2j What is its speed after 5 seconds? COMMON MISCONCEPTIONS/EXAMINER REPORT QUOTES Encourage students to draw diagrams to help geometrical thinking when answering vector questions. Stress the importance of reading the question carefully and giving answers in the correct way, for example coordinates or column vectors may be requested. Emphasise the importance of good notation. Students do not always understand that represents the square of the length AP. NOTES Please note that vector equations of straight lines and the scalar product are on this specifications. not
TEACHING POINTS Begin by showing some 3D coordinates on x, y, z axes. (Graph drawing packages are very useful here, especially if you can turn the grid to view from different positions.) Consider a cuboid (2 by 3 by 4), with one corner at the origin. Ask the class to write down the coordinates of all the vertices. Remind students of 2D work and extend to 3D column vectors, orthogonal unit vectors i, j, k and position vectors. Write all the vectors from the ‘origin’ corner of the cuboid as position vectors (e.g. OA = a, etc.) Calculate the magnitude of these vectors as for example. Extend this idea to calculating the distance d between two points (x1, y1, z1) and (x2, y2, z2) using d 2 = (x1 – x2)2 + (y1 – y2)2 + (z1 – z2)2 Extend the ideas of vector addition and subtraction to 3D: = b – a. Cover the triangle and parallelogram laws of addition, as well as demonstrating parallel vectors. Show how to find a unit vector in the direction of a, and make sure students are familiar with the notationa (extended to 3D). Use vectors to solve problems in pure mathematics and discuss the 3D geometrical interpretations of solutions. OPPORTUNITIES FOR REASONING/PROBLEM SOLVING Link examples to mechanics (kinematics and forces). For example, consider questions such as: The velocity of an object is given by vector v = 3ti + t2j + 4k What is its speed after 5 seconds? COMMON MISCONCEPTIONS/EXAMINER REPORT QUOTES Encourage students to draw diagrams to help geometrical thinking when answering vector questions. Stress the importance of reading the question carefully and giving answers in the correct way, for example coordinates or column vectors may be requested. Emphasise the importance of good notation. Students do not always understand that represents the square of the length AP. NOTES Please note that vector equations of straight lines and the scalar product are not on this specifications.
A_level_Maths_2_years_SoW.docx