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Vectors (I)

Geometry · Vectors (I)

Syllabus tag: KCSE | Mathematics | Form 2 | Topic 20 Vectors (I)

Lesson objectives

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

  • Define a vector and distinguish it from a scalar.
  • Use vector notation and represent vectors geometrically.
  • Identify equivalent vectors and add vectors.
  • Multiply a vector by a scalar and find the magnitude of a vector.

Vectors (I)

A vector has both size and direction. A scalar has size only.

a) The distinction

Scalar or vector Scalar or vector scalar vector distance displacement speed velocity mass weight size only size and direction

Distance is a scalar; displacement is a vector.

Speed is a scalar; velocity is a vector.

Mass is a scalar; weight is a vector, since it is a force acting downwards.

b) Notation

A vector may be written as AB with an arrow above, meaning from A to B.

In print it is often written in bold, as a, or underlined when handwritten.

Underline your vectors in written work. A vector written as a plain letter is easily read as a scalar.

c) Column vectors

A vector is written as a column. The top number is the movement in x, the bottom the movement in y.

So (3, 1) means 3 across and 1 up.

Negative components mean movement left or down.

d) Equivalent vectors

Two vectors are equal if they have the same magnitude and the same direction.

Position does not matter. A vector drawn anywhere on the page is the same vector, provided its length and direction match.

This is why vectors can be moved around freely when adding.

e) Adding vectors

Adding vectors Adding vectors a = (3, 1) as a column vector b = (2, 4) another a + b = (3+2, 1+4) add the components a + b = (5, 5) the resultant

Add the components separately: x with x, and y with y.

Geometrically, place the second vector's tail at the first vector's head. The resultant runs from the start of the first to the end of the second.

This is the triangle law. Going A to B then B to C is the same as going straight from A to C.

f) Negative vectors and subtraction

−a has the same magnitude as a but the opposite direction.

To subtract, add the negative: a − b = a + (−b).

In components, subtract them separately.

g) Multiplying by a scalar

Multiplying a vector by a scalar changes its length, not its direction.

So 3a is three times as long as a, pointing the same way.

A negative scalar reverses the direction as well. So −2a is twice as long and points the opposite way.

If two vectors are scalar multiples of each other, they are parallel.

h) Magnitude

Magnitude of a vector Magnitude of a vector v = (3, 4) the components magnitude = √(3² + 4²) Pythagoras magnitude = √25 add the squares magnitude = 5 the length of the vector

The magnitude is the vector's length, written |a|.

By Pythagoras, |a| = √(x² + y²).

Magnitude is always positive, since it is a length. A vector of (−3, −4) still has magnitude 5.

i) Where this is used

Forces in physics. Navigation, where course and wind combine. Displacement problems. Computer graphics.

Words to know

  • Vector -- a quantity with both magnitude and direction.
  • Scalar -- a quantity with magnitude only.
  • Resultant -- the single vector equivalent to two or more combined.
  • Magnitude -- the length of a vector, always positive.
  • Position vector -- the vector from the origin to a point.

:::checkpoint Check yourself

  1. Give two examples of scalars and two of vectors.
  2. Add the vectors (4, −2) and (1, 5).
  3. Find the magnitude of the vector (6, 8).
  4. What does multiplying a vector by −3 do to it? :::

Bridge to practice

The exercises begin with the vector-scalar distinction and notation, move through equality, addition and scalar multiplication, and finish with magnitude, position vectors and collinearity. For every vector between two points, write down which point is the destination before subtracting.

Check yourselfPractise Vectors (I) — 10 questions →