Examples and practice
Vectors: magnitude, direction and products
Back to lessonPractice section 1 · Questions 1–6
Walk a fixed distance in a vector’s direction · Find a position vector, its modulus and scaled directions · +4 more
- Walk a fixed distance in a vector’s direction
- Find a position vector, its modulus and scaled directions
- Prove the triangle midpoint theorem with vectors
- Prove collinearity from two proportional displacements
- Distinguish collinear points from a failed scalar relation
- Calculate displacements and recognise a parallelogram
Practice section 2 · Questions 7–12
Distinguish internal division from a line produced · Locate internal and external division points in coordinates · +4 more
Practice section 3 · Questions 13–18
Find a parameter from perpendicular vectors · Combine a known angle and a right-angle condition · +4 more
Practice section 4 · Questions 19–24
Prove the directional-cosine identity for a position vector · Separate projection length, parallel vector and perpendicular vector · +4 more
- Prove the directional-cosine identity for a position vector
- Separate projection length, parallel vector and perpendicular vector
- Keep a negative signed component in a vector projection
- Use a zero cross product to prove parallelism
- Find and verify a common perpendicular vector
- Calculate two cross products in the stated order
Practice section 5 · Questions 25–30
Relate parallelogram and triangle areas · Derive triangle area from three position vectors · +4 more
- Relate parallelogram and triangle areas
- Derive triangle area from three position vectors
- Give a direction vector the required magnitude
- Use equal vectors and diagonal lengths to identify a quadrilateral
- Apply the section formula and preserve vertex order
- Find a perpendicular foot and use it as a triangle height
Practice section 6 · Questions 31–36
Expand dot products using magnitudes and an angle · Prove a projection formula in two ways · +4 more
Practice section 7 · Questions 37–42
Solve perpendicularity conditions and compare two area methods · Connect a cross-product relation to areas and a nonzero linear combination · +4 more
- Solve perpendicularity conditions and compare two area methods
- Connect a cross-product relation to areas and a nonzero linear combination
- Use one displacement for four vector tasks
- Find a displacement and prove a line intersection ratio
- Recover a dot product, projection and a parameterised area
- Find a perpendicular foot on a parameterised line
Practice section 8 · Questions 43–48
Distinguish four independent parameter cases · Extend a segment and use two vectors based at the angle vertex · +4 more
- Distinguish four independent parameter cases
- Extend a segment and use two vectors based at the angle vertex
- Expand a cross product for an area ratio
- Use an ordered parallelogram and a norm constraint
- Remove a parallel component from an area calculation
- Prove orthogonality and explain the zero-vector case
Practice section 9 · Questions 49–52
Use a projection foot to find a pyramid base area · Relate external division, area ratios and a unit-vector projection · +2 more