Vectors

What vectors covers

Quantities with a direction attached. Arrows add tip to tail, position vectors turn geometry into arithmetic, and mechanics is built on them. Two lessons, and they stay with components, magnitudes and scalar multiples: the scalar product and the vector equation of a line belong to Further vectors in 9FM0 rather than here.

The main ideas

• Column and i-j notation, and converting between components and magnitude-direction form.
• Addition by the triangle and parallelogram laws, scalar multiplication, and parallel vectors recognised as scalar multiples.
• The unit vector in a given direction, and scaling it to any length a question asks for.
• Position vectors, the rule that AB is b minus a, and the distance between two points.
• The third component: i, j and k notation, with magnitude from the sum of three squares.
• Geometry done by distance and scalar multiple alone: midpoints, collinear points, the fourth corner of a parallelogram, and triangles shown to be isosceles or right angled.

The results it turns on

|a| = √(x² + y²), and √(x² + y² + z²) in three dimensions

the magnitude of a vector, by Pythagoras on its components

AB$\overrightarrow{AB}$vec AB = b − a

the vector from A to B, destination minus start

the unit vector in the direction of a is a/|a|

the direction of a, rescaled to length one

a = kb for some scalar k

the test for parallel vectors, and for collinearity through a shared point

the midpoint of AB has position vector ½(a + b)

the position vector of the midpoint of a line segment

Every result the vectors notes display is collected, with its conditions, on the vectors formulae page.

Where it usually goes wrong

• A direction is meaningless without a stated reference direction, so quote the angle against a named axis or bearing rather than leaving it bare.
• Collinearity is parallelism plus a shared point. Both the scalar and the common point have to be named for the argument to be complete.
• In three dimensions the middle component is where sign slips gather, so keep the working in a column rather than running it along a line.

Where to start

Two dimensions first, then three, since the second lesson adds one slot and changes nothing else. Revisit the unit before starting Mechanics, where forces in i-j notation and the magnitude of a resultant are used from the first lesson onwards.