Vectors

What vectors covers

Quantities with a direction attached. Arrows add tip to tail, position vectors turn geometry into arithmetic, and mechanics waits on the other side. 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

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.