Vectors in two and three dimensions
Build and connect the representations and methods in vectors in two and three dimensions.
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Before you begin
Build and connect the representations and methods in vectors in two and three dimensions.
Learning goals
- Add vectors by ordered components
- Subtract vectors by ordered components
- Multiply a vector by a scalar
- Calculate the magnitude of a vector
- Produce a unit vector in a given direction
- Locate a point from a displacement parameter
- Find the directed displacement between two points
- Calculate the distance between two points
- Test all coordinates for point collinearity
- Locate an internally or externally dividing point from its ratio
- Explain vector geometry, position, displacement and ratio relationships
- Calculate a scalar product
- Calculate an ordered vector product
- Calculate the angle between directed vectors
- Find a signed scalar component along a direction
- Find the parallel vector component along a direction
- Find the perpendicular component's magnitude
- Justify the geometrical meanings and properties of vector products
- Find the parallel component's magnitude
- Justify spatial vector relationships and parameter conditions
- Produce a vector equation of a spatial line
- Produce independent Cartesian equalities for a spatial line
- Produce a vector equation of a plane
- Produce a Cartesian equation of a plane
- Distinguish coincident, parallel, intersecting and skew lines
- Classify a line's relationship with a plane
- Distinguish coincident, parallel and intersecting planes
- Produce the intersection line of two planes
- Find the unique intersection point of a line and a plane
- Find the foot of a perpendicular from a point to a line
- Find the perpendicular distance from a point to a line
- Find the foot of a perpendicular from a point to a plane
- Find the perpendicular distance from a point to a plane
- Find the smaller angle between spatial lines
- Find the smaller angle between planes
- Find the angle between a line and a plane
Syllabus statements covered
- Include: vector addition, subtraction, scalar multiplication and geometrical interpretations; position, displacement and direction vectors; magnitude; unit vectors; distance between points; collinearity; ratio theorem in geometrical applications.
- Include: scalar and vector products and properties; angle between vectors; geometrical meanings of |a · n̂| and |a × n̂| for unit vector n̂. Exclude triple products a · (b × c) and a × (b × c).
- Include: vector and Cartesian equations of lines and planes; foot of perpendicular and distance from point to line or plane; angles between lines, line and plane, or planes; relationships between coplanar or skew lines, line and plane, and planes. Exclude shortest distance and common perpendicular between two skew lines.
- Include: vector addition, subtraction, scalar multiplication and geometrical interpretations; position, displacement and direction vectors; magnitude; unit vectors; distance between points; collinearity; ratio theorem in geometrical applications.
- Include: scalar and vector products and properties; angle between vectors; geometrical meanings of |a · n̂| and |a × n̂| for unit vector n̂. Exclude triple products a · (b × c) and a × (b × c).
- Include: vector and Cartesian equations of lines and planes; foot of perpendicular and distance from point to line or plane; angles between lines, line and plane, or planes; relationships between coplanar or skew lines, line and plane, and planes. Exclude shortest distance and common perpendicular between two skew lines.
Lessons
Work through them in order.
About lesson progress
Lesson marks record what you have studied. Checks show understanding separately. Guest marks come from this device.
- Vector geometry and productsBuild vector ratios, signed projections and products from their geometrical meaning.
- Lines, planes and intersectionsBuild line and plane equations, verify intersections and interpret parameter restrictions.
- Perpendicular feet, distances and anglesDerive perpendicular feet and distances, then choose the correct line or plane angle.