About this topic
Before you begin
Use vector ratios and products to form lines and planes, verify intersections, and calculate perpendicular distances and angles.
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.
Choose a lesson
Work through them in order.
- Not startedVector geometry and productsNextBuild vector ratios, signed projections and products from their geometrical meaning.
- Not startedLines, planes and intersectionsNextBuild line and plane equations, verify intersections and interpret parameter restrictions.
- Not startedPerpendicular feet, distances and anglesNextDerive perpendicular feet and distances, then choose the correct line or plane angle.