Section 12-1

Geometric Representation of Vectors

Vectors

Vectors are quantities that are described by a direction and a magnitude (size). A force would be an example of a vector quantity because to describe a force, you must specify the direction in which it acts and its strength. Velocity is another example of a vector.

Vectors

The velocities of two airplanes each heading northeast at 700 knots are represented by the arrows u and v in the diagram on p. 419. We write u = v to indicate that both planes have the same velocity even though the two arrows are different.

Vectors

In general, any two arrows with the same length and the same direction represent the same vector. The diagram on p. 419 shows a third airplane with speed 700 knots, but because it is heading in a different direction, its velocity vector w does not equal either u or v.

Magnitude

The magnitude of a vector v (also called the absolute value of v) is denoted |v|.

Addition of Vectors

If a vector v is pictured by an arrow from point A to point B, then it is customary to write v = . Since the result of moving an object first from A to B and then from B to C is the same as moving the object directly from A to C, it is natural to write . We say that is the vector sum of and .

AB

ACBCAB

BCABAC

Addition of Vectors

The addition of two vectors is a commutative operation. In other words, the order in which the vectors are added does not make any difference. You can see this in the diagrams on p. 420 where the red arrows denote a + b and b + a having the same length and direction.

Addition of Vectors

If the two diagrams on p. 420 are moved together, a parallelogram is formed. This suggests that another way to add a and b is to draw a parallelogram OACB with sides = a and = b. The diagonal of the parallelogram is the sum.

This method is frequently used in physics problems involving forces that are combined.

OBOA OC

Vector Subtraction

The negative of a vector v, denoted –v, has the same length as v but the opposite direction. The sum of v and –v is the zero vector 0.

It is best thought of as a point. v + (-v) = 0

Vector Subtraction

Vectors can be subtracted as well as added.

v – w means v + (-w).

Multiples of a Vector The vector sum v + v is abbreviated

as 2v. Likewise, v + v + v = 3v. The diagram on p. 421 shows that the arrows representing 2v and 3v have the same direction as the arrow representing v, but that they are two and three times as long.

Multiples of a Vector

In general if k is a positive real number, then kv is the vector with the same direction as v but with an absolute value k times as large. If k < 0, then kv has the same direction as –v and has an absolute value |k| times as large. If k ≠ 0, then is defined to be equal to the vector . k

v

vk

1

Scalars

When working with vectors, it is customary to refer to real numbers as scalars.

Scalar Multiplication

When this is done, the operation of multiplying a vector v by a scalar k is called scalar multiplication. This operation has the following properties. If v and w are vectors and k and m are scalars, then:

k(v + w) = kv + kw

(k + m)v = kv + mv

k(mv) = (km)v Associative law

Distributive laws

True or False?

C

Complete the statement.

C

Homework: p. 423-424 1-9 odd

Homework: p. 423-424 1-9 odd

Homework: p. 423-424 1-9 odd

Homework: p. 423-424 1-9 odd

Homework: p. 423-424 1-9 odd

9. A ship travels 200 km west from port and then 240 km due south before it is disabled. Illustrate this in a vector diagram. Use trigonometry to find the course that a rescue ship must take from port in order to reach the disabled ship.