Vectors worksheets with answers

Introduces the scalar product, derived from a·b = |a||b| cos θ, and uses it to find the angle between two vectors. Students also use a·b = 0 to show that two vectors are perpendicular, or to make them so.

AQA 7366/7367

F3, F4

Edexcel 9FM0

Core Pure 6.3, 6.4

Writes the vector equation of a line in 3D, r = a + λd, including the line through two points. Students check whether a point lies on a line, and recognise the same line written in different ways, and parallel lines.

AQA 7366/7367

F1

Edexcel 9FM0

Core Pure 6.1

Derives the Cartesian form of a line from its vector equation and converts between the two forms, including when a direction component is zero. Students use the Cartesian form to find points, unknown coordinates, where a line crosses a plane, and parallel lines.

AQA 7366/7367

F1

Edexcel 9FM0

Core Pure 6.1

Finds the acute angle between two lines from their direction vectors, whichever form the lines are given in. Students also use the scalar product for perpendicular lines and unknowns, and find the angle a line makes with an axis.

AQA 7366/7367

F1, F4

Edexcel 9FM0

Core Pure 6.3, 6.4

Decides whether two lines in 3D are parallel, intersect or are skew, using separate parameters and checking the third component. Students find points of intersection, unknowns that make lines meet, and the angle between lines where they meet.

AQA 7366/7367

F1

Edexcel 9FM0

Core Pure 6.5

Finds shortest distances using perpendiculars: from a point to a line (through the foot of the perpendicular), between two parallel lines, and between two skew lines. Students see why the shortest distance is always along a perpendicular.

AQA 7366/7367

F1, F5

Edexcel 9FM0

Core Pure 6.5