VECTOR ADDITION IN THE PLANE

Vectors in the plane are given by arrows based at the origin. We denote by the single letter A the vector from the origin O = (0,0) to the point (a,b).

To add two vectors A = (a,b) and C = (c,d), we add their coordinates, so A + C = (a,b) + (c,d) = (a+c,b+d).

Note that the slope of the line containing (a,b) and (a+c,b+d) is the same as the slope of the line containing (0,0) and (c,d) so these two lines are parallel. Simimilarly the slope of the line containing (c,d) and (a+c,b+d) is the same as the slope of the line containing (0,0) and (a,b) so these two lines are also parallel. It follows that the quadrilateral from O to A to A+C to C and back to O has opposite sides parallel so it is a parallelogram. The point A+C is therefore the intersection of the line through A parallel to the line along C and the line through C parallel to the line along A. The definition of vector addition is often called "the parallelogram law".

Note that if A and C are themselves parallel, lying along a line through the origin, then A+C also lies on this line.


[D]

SCALAR MULTIPLICATION IN THE PLANE


Multiplication of a vector A by the real number r is defined by rA = r(a,b) = (ra, rb).

Geometrically rA represents a stretching of A by a scale factor r if r > 1 and a shrinking by a scale factor r is 0 < r < 1. We speak of the number r as a "scalar" and rA as a "scalar multiple of A". Note that 1A = 1(a,b) = (1a,1b) = (a,b) = A and 0A = O(a,b) = (0a,0b) = (0,0) = O. Multiplying A by -1 produces a vector -1A = -1(a,b) = ((-1)a,(-1)b) = (-a,-b), and when we add this vector to A, we get A + (-1)A = (a,b) + (-a,-b) = (0,0) = O. Thus (-1)A is the "additive inverse of A" denoted -A. Similarly if 0 < r, then (-r)A = -(rA), the additive inverse of rA.

It follows that the collection of all vectors tA where t is a real number is "the line along A through the origin".


[D]

VECTOR EQUATIONS FOR LINES IN THE PLANE

The line through C parallel to the line along A is the collection of points C + tA for all real t.

Similarly the line through A parallel to the line alonge C is the collection A + sC for real scalars s. If A and C are not themselves parallel, then the point A+C is the intersection of the line through A parallel to the line along C and the line through C parallel to the line along A, obtained when t and s are both equal to 1.

  [D]
[D]

LENGTH, ANGLE, AND DOT PRODUCT IN THE PLANE

By the Pythagorean theorem, the "length" of the vector from O = (0,0) to A = (a,b) is √(a2 + b2), denoted |A|.

Note that |A| >= 0 for all A, and |A| = 0 if and only if A = O.

Furthermore |cA| = |c||A| where |c| denotes the absolute value of the scalar c,

Also the distance from A to C is given by the length of the vector C-A, and |C-A| < |C| + |A|.  This is called the "triangle inequality".

Each of these three properties can be proved by using coordinate.

Condition for Perpendicularity:  By the Pythagorean Theorem, the vectors A and C are the legs of a right triangle if |A|2 + |C|2 = |C-A|2, which in coordinates becomes a2 + b2 + c2 + d2 = (c-a)2 + (d-a)2 = c2 - 2ac + a2 + d2 -2db + b2.  This condition is then equivalent to ac + bd = 0.  We define the "dot product" or "inner product" of the vectors A = (a,b) and C = (c,d) to be ac + bd = A*C.


[D]

Note that A*(C+E) = A*C + A*E and (tA)*C = t(A*C).  Also A*C = C*A for any two vectors A and C, and A*A = |A|2, which equals zero if and only if A = O.  Note also that the dot product of any vector with the zero vector is 0, so we say that O is perpendicular to all vectors.  Note that (C-A)*(C-A) = C*(C-A) + (-A)*(C-A) = C*C - C*A - A*C + A*A = C*C - 2A*C + A*A.

Exercise:  Find the value of t such that the vector A + tC is perpendicular to C, where C is a non-zero vector.

If A and C are non-zero vectors that make an angle θ at the origin, then (|A|sin(θ))2 = |C-A|2 - (|C|-|A|cos(θ))2 = |C-A|2 - |C|2 + 2|A||C|cos(θ) - (|A|cos(θ))2, from which it follows that 2|A||C|cos(θ) = |A|2 + |C|2 - |C-A|2 = A*A + C*C - (C-A)*(C-A) = 2A*C, by the above calculation.  Therefore A*C = |A||C|cos(θ), and cos(θ) = A*C/(|A||C|).


[D]

Exercise:  Find the cosine of the angle between A = (cos(a),sin(a)) and C = (cos(c),sin(c)) where 0 < a < c < π/2.

Exercise:  Show that the area of the triangle with sides A and C equals (1/2)|A||C|sin(θ) = (1/2)√((A*A)(C*C)-(A*C)2) = (1/2)|ad - bc| Then observe that the area of the parallelogram with vertices O, A, A+C, and C is equal to |ad - bc|.