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Linear Algebra · Book or PDF
A First Course in Linear Algebra
A.G. Hamilton · Study material generated by AI from this edition. Not affiliated with the publisher.
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A.G. Hamilton · Study material generated by AI from this edition. Not affiliated with the publisher.
Chapter 9 · pp. 96–113
Column vectors can describe where one point sits relative to another, so algebra and geometry can be used together. This chapter shows how adding, scaling and subtracting vectors match geometric moves. It then uses these ideas to prove results about dividing a line segment, finding the centre of a triangle, and measuring length in three dimensions.
Once coordinate axes are chosen, you can describe the position of a point Q relative to a point P with a column vector, written . To find it, move from P parallel to the axes until you reach Q. Each number in the column records one of these moves. Its size is the length of the move. Its sign is positive if the move goes the same way as that axis and negative if it goes the opposite way (pp. 96, 98).
Example: going 3 units along OX and 2 units against OY gives . In three dimensions there is a third number for the third axis.
Three facts link points and vectors (p. 98):
Adding column vectors adds the horizontal moves and the vertical moves separately. This matches the geometric idea of making one move and then another. The result is the Triangle Law: for any points P, Q, R,
Example: moves and combine to , the single move from the start to the finish (pp. 99–100).
The Parallelogram Law follows from this. Suppose PQRS is a parallelogram. Its opposite sides QR and PS are parallel, equal in length and point the same way, so . Put this into the Triangle Law and you get
So the diagonal from P is the sum of the two sides that start at P (pp. 99–100).
These laws do not depend on which axes you chose. They are geometric facts that vectors express neatly (p. 102).
When the starting point is the origin O, the vector is the position vector of A, written . Its components are exactly the coordinates of A, because the construction lines run along the axes. For example, the point has position vector . Each point has exactly one position vector (p. 102).
Multiplying a vector by a number (a scalar) changes its length by the factor .
The zero vector comes from no pair of distinct points. You can still think of it as the position vector of the origin itself (p. 104).
The Triangle Law gives for a triangle PAB. Rearranging gives . If you take P to be the origin, this becomes
In words: to get the vector from A to B, take the position vector of the end and subtract the position vector of the start. Example: for and , (pp. 105–106). In a parallelogram, one diagonal is the sum of the two sides from a corner, and the other diagonal is their difference (p. 105).
A point P on line AB (possibly beyond the segment) divides AB in the ratio when . If one of the numbers is negative, then and point opposite ways, and P lies outside the segment (p. 106).
To derive the formula, write . Since and , we get
Solving for gives the Section Formula (pp. 106–108):
Notice that the ratio's first number goes with B. The point is pulled toward the end that has the bigger weight.
For the midpoint, , so (p. 108).
Example 1: for , and ratio , .
Example 2: for , and ratio , . This point is outside the segment, beyond A.
A median joins a vertex to the midpoint of the opposite side. Let the triangle ABC have position vectors , and let L, M, N be the midpoints of BC, CA, AB. Then L has position vector , and M and N follow the same pattern (pp. 107–108).
Use the Section Formula with ratio on AL. This gives
The same calculation on BM and on CN gives the very same point. So this point lies on all three medians. It is the centroid, and it divides each median in the ratio (pp. 107–108).
Example: for , , , the centroid is .
The proof never used particular coordinates. It is a purely geometric result obtained with vector algebra (p. 108).
In three dimensions, the points , , have position vectors , , . These are the standard basis vectors. For a point , build a box from P down to the axes. The Parallelogram Law applied twice gives
Every 3-vector can be written this way in exactly one way, and are its components along the axes (pp. 109–110).
The length of a vector is the length of any segment that represents it. For it is the length of OP:
The vectors each have length 1 (p. 110).
A unit vector has length 1. Given a non-zero vector , dividing by its length gives a unit vector in the same direction: . Example: has length , so is a unit vector (pp. 109–110).