Vectors
Senior 2 · Mathematics
Vectors
Lesson 2 of 4
18 min
Two vectors that describe the same move are equal, wherever they are drawn. That simple idea is powerful. It lets you prove that four points form a parallelogram, or find a missing corner of a plot of land, with nothing more than subtraction. In this lesson you also meet the special vectors that every later calculation uses: the null vector, position vectors and unit vectors.
The null vector (or zero vector) is 00, written 0. It is the "move" from a point to itself, so AA = 0. Its magnitude is 0 and it has no direction.
The position vector of a point P is the vector OP from the origin O(0, 0) to P. If P is (x, y), then OP = xy: the same numbers as the coordinates. For example, the position vector of P(−3, 7) is −37.
This explains the rule from the last lesson: AB = OB − OA.
A unit vector is a vector whose magnitude is 1. Two unit vectors are so useful that they have their own names:
i = 10 (one unit right) and j = 01 (one unit up).
Every column vector can be built from them. For example 3−2 = 3i − 2j: three steps of i and two steps against j.
Two vectors are equal when they have the same magnitude, the same direction and the same sense. On the Cartesian plane this simply means they have the same components. Where the vector is drawn does not matter.
Worked example 1
A(1, 1), B(4, 2), C(5, 5) and D(2, 4). Show that ABCD is a parallelogram.
AB = 4 − 12 − 1 = 31 and DC = 5 − 25 − 4 = 31.
So AB = DC: the sides AB and DC are parallel and equal in length. A quadrilateral with one pair of opposite sides parallel and equal is a parallelogram.
Figure 2. The arrows AB and DC are drawn in different places but describe the same move, so they are equal vectors. Diagram: Ganzaa.
The opposite of a is −a: same magnitude and direction, opposite sense. Change the sign of both components: the opposite of 31 is −3−1. In particular BA = −AB.
Worked example 2
Find x and y if x + 25 = 7y − 1.
Equal vectors have equal components: x + 2 = 7, so x = 5; and 5 = y − 1, so y = 6.
Worked example 3
A(0, 1), B(4, 2) and C(6, 5) are three corners of parallelogram ABCD. Find D.
In ABCD, AD = BC = 6 − 45 − 2 = 23. So D = (0 + 2, 1 + 3) = (2, 4).
Check: DC = 6 − 25 − 4 = 41 and AB = 41. They are equal, as they must be.
Tip
Walk round the shape. In parallelogram ABCD the letters go round in order, so the equal pairs are AB = DC and AD = BC. Writing AB = CD is the classic slip: CD points the opposite way.
Write the position vector of P(−3, 7).
Solution: −37.
Write 4−5 in terms of i and j.
Solution: 4i − 5j.
Are u = 2−6 and v = −26 equal?
Solution: No. They have the same magnitude and direction but opposite sense: v = −u.
Solve 2ab + 3 = 10−1.
Solution: 2a = 10, so a = 5; b + 3 = −1, so b = −4.
A(−1, 0), B(3, 3) and C(5, 1). Find D so that ABCD is a parallelogram.
Solution: BC = 2−2, so D = A + BC = ( − 1 + 2, 0 − 2) = (1, −2). Check: DC = 5 − 11 − ( − 2) = 43 = AB.
Remember
Sources and credits
The Rwandan examples, practice items and exercises are written by Ganzaa.
