Vectors and Three-Dimensional Geometry

About this unit

Vectors are quantities with both magnitude and direction, such as displacement, velocity and force. This unit teaches you to represent vectors in components along the unit vectors i, j and k, to add and scale them, to divide line segments in a given ratio, and to multiply vectors in two ways: the dot (scalar) product and the cross (vector) product. These tools are then used in three-dimensional geometry to write equations of lines in space, find the angle between two lines and find the shortest distance between skew lines.

Key ideas

• A vector a = a₁i + a₂j + a₃k has magnitude √(a₁² + a₂² + a₃²).
• Direction cosines l, m, n satisfy l² + m² + n² = 1; direction ratios are any numbers proportional to them.
• Section formula gives the position vector of a point dividing a segment in a ratio.
• a · b = |a||b| cos θ is a number; a × b is a vector perpendicular to both a and b.
• A line through a point with position vector a, parallel to b, is r = a + λb.
• Skew lines are neither parallel nor intersecting; their shortest distance uses the cross product.

How to prepare

Memorise the formulas for dot and cross products, the section formula and the shortest-distance formula. Practise converting between vector and Cartesian forms of a line. Questions are usually short computations, so accuracy with signs in determinants is essential.