Vectors (II)

Syllabus tag: KCSE | Mathematics | Form 3 | Topic 15 Vectors (II)

Lesson objectives

By the end of this topic, you should be able to:

• Locate a point in two and three dimensional coordinate systems.
• Represent vectors as column and position vectors in three dimensions.
• Express vectors in terms of the unit vectors i, j and k.
• Apply the ratio theorem and use vectors to prove geometric results.

Vectors (II)

Form 2 vectors worked in two dimensions. This topic extends to three, and uses vectors to prove geometric results.

a) Three-dimensional coordinates

A point in space needs three coordinates, written (x, y, z).

The z-axis is perpendicular to both the x and y axes.

b) Position vectors

The position vector of a point is the vector from the origin to that point.

The position vector of A is written OA, or sometimes a.

Any vector between two points follows: AB = OB − OA.

Read that as "to B, from A". Getting it backwards gives BA instead, which points the wrong way.

c) Unit vectors

i, j and k are vectors of length 1 along the x, y and z axes.

Any vector can be written in terms of them.

So the column vector (2, 3, 6) is the same as 2i + 3j + 6k. The two notations are interchangeable.

d) Adding and scaling

Add component by component, exactly as in two dimensions.

Multiplying by a scalar multiplies every component.

e) Magnitude in three dimensions

|v| = √(x² + y² + z²).

This is Pythagoras applied twice: once in the base plane, then again with the height.

f) Parallel vectors and collinear points

Two vectors are parallel if one is a scalar multiple of the other.

Three points are collinear if one joining vector is a scalar multiple of another. The two must also share a common point.

The shared point matters. Parallel alone gives parallel lines, not the same line.

g) The ratio theorem

Let R divide AB in the ratio m : n. Then OR = (nOA + mOB) / (m + n).

Note the swap: n multiplies OA, and m multiplies OB.

A point nearer A carries more of OA, and being nearer A means m is small. So the smaller ratio number pairs with the further point.

For the midpoint, m = n, giving OR = (OA + OB)/2, which is the average as expected.

h) Proving geometric results

Express every vector in the figure in terms of two chosen base vectors.

To prove lines are parallel, show one vector is a scalar multiple of the other.

To prove points are collinear, do the same and show a shared point.

To find where two lines cross, write the point two ways using unknown scalars. Then equate the coefficients of each base vector, giving simultaneous equations.

Equating coefficients works because the base vectors are independent. Neither can be made from the other, so their coefficients must match separately.

i) Where this is used

Structural mechanics. Navigation in three dimensions. Computer graphics. Proving results in geometry without coordinates.

Words to know

• Unit vector -- a vector of magnitude 1, such as i, j or k.
• Position vector -- the vector from the origin to a point.
• Ratio theorem -- the formula locating a point dividing a line in a given ratio.
• Collinear -- lying on the same straight line.
• Base vectors -- two non-parallel vectors used to express all others in a figure.

:::checkpoint Check yourself

1. Write the vector (3, −1, 5) in i, j, k form.
2. Find the magnitude of 6i + 2j + 3k.
3. If AB = OB − OA, what is BA?
4. Find the position vector of the midpoint of A(2, 4) and B(6, 10). :::

Bridge to practice

The exercises begin with three-dimensional coordinates and i, j, k notation, move through magnitude and the ratio theorem, and finish with collinearity and vector proofs. When applying the ratio theorem, write m and n above the formula first so the swap is visible.