


Vectors
Often used to represent velocity, distance traveled, or
force, vectors are line segments defined by two properties: length
and direction.
Graphically, a vector is shown as a line segment with
an arrow on one end, to show its direction:
In the preceding figure, vector u,
which can also be written , has initial point
(–2, –2), and terminal point (6, 4). When you see a vector equated
to a coordinate pair, say u =
(8, 6), like in the figure above, the first coordinate is called
the xcomponent of the vector and the second coordinate
is the ycomponent.
The length of a vector, also known as its magnitude, equals
the square root of the sum of the squares of its components. Basically,
it’s a restatement of the Pythagorean theorem. The length of the
vector above, for example, is = 10.
The length of a vector can be found if you know its initial
point and terminal point, or if you know the components of the vector.
Adding and Subtracting Vectors
To add or subtract vectors, just add or subtract their
respective components. For example, if a = (2, 7) and b =
(–5, 2), a + b = (2 +
–5, 7 + 2) = (–3, 9). In order to graph this, you first place the
second vector so that its tail starts at the tip of the first vector.
Then close the triangle by drawing a vector from the tail of the
first vector to the tip of the second vector. It looks like this:
Subtracting vectors is the same process, except that you
reverse the direction of the subtracted vector: a – b = a +
(–b) = (2 – (–5), 7 –
2) = (7, 5).
Multiplying by a Scalar
Occasionally, the Math IIC will ask you to multiply a
vector by a scalar, which is a number that has magnitude but no
direction. To answer this type of question, just multiply each component
of the vector by the scalar. If u = (3,
4), cu = (3c,
4c).
Try an example problem:

It’s easy—just multiply and then subtract: 3u –
2v = 3(1, –5) – 2(4, 2) = (3, –15) – (8, 4) = (–5, –19).
Calculating the length, sum, difference, or product of
vectors and scalars is the extent of what the Math IIC will ask
you on this subject.
