8.1 and 8.2 answers

8.3: Vectors

February 9, 2009

Objectives

• Learn basic concepts about vectors.• Perform operations on vectors.• Learn and apply the dot product.• Use vectors to calculate work.

The basics: Scalars vs. Vectors

• Scalars are quantities which are fully described by a magnitude (or numerical value) alone.

• Vectors are quantities which are fully described by both a magnitude and a direction.

More about vectors

• A vector quantity can be represented by a directed line segment.

• Vectors are equal if they have the same direction and magnitude.

• They are usually represented by a letter in boldface type, such as a, b, v, or F.

Describing vectors

If the initial point is placed on the origin, then its terminal point (a1, a2) can bed used to determine v.

a1 is the horizontal component.

a2 is the vertical component.

Magnitude of a vector

If v = <a1, a2>, then its magnitude (or length) is:

||v|| =√ (a12 + a2

2)

If ||v|| = 1, then v is a unit vector.

Vector Addition

If a = < a1, a2> and b = < b1, b2>, the sum of a and b is…

a+b = < a1, a2> + < b1, b2>

= < a1+ b1, a1+ b2>,

Vector Subtraction

If a = < a1, a2> and b = < b1, b2>, the difference of a and b is…

a - b = < a1, a2> - < b1, b2>

= < a1- b1, a1- b2>,

Scalar multiplication

If v = < v1, v2> and k is a real number, then the scalar product kv is

kv= k < v1, v2>

= < kv1, kv2>

Dot product

If a = < a1, a2> and b = < b1, b2>, the dot product of a and b is…

a • b = a1b1 + a2b2

Angle between two vectors

If a and b are two non-zero vectors, then the angle θ between a and b is given by

θ = cos -1 (a • b)

(||a|| ||b||)

Vectors are perpendicular if and only if a • b =0.

Work

If a constant force F is applied to an object that moves along a vector D, then the work W done is

W=F • D

assignment

• Pages 687-8• 25-34• 51-62