VectorsDay 2

Scalar Multiplication

• A vector can be multiplied by a real number• Multiplying a vector by a positive number changes its size, but not its

direction.• Multiplying a vector by a negative number changes its direction and its size

(unless it is multiplied by -1)

• The multiplication of a scalar, k, and a vector, v, is denoted as kv• A scalar “scales” the size of the vector.

Adding vectors – “The Triangle Method”

• The process of geometrically adding two vectors is as follows:• Given vector v and vector u

1) Draw vector v2) At the terminal point of v, draw vector u3) Draw the resultant vector (r) from the initial point of v to the terminal point

of u

Examples

• 1. v + u • 2. u + v

u

v

r

u

v

r

Look!!!

u

v

r

u

v

r

Example: Subtraction

• 4. u - v

u

v

r

v

Adding vectors in component form

Given v = and w = 1. Geometrically add the vectors

• Find the component form of the resultant vector.

Scalar Multiplication and Component Form

• Given a vector v = and a scalar k,kv= =

Examples

• Given v = and u = , find each of the following: 1. v + u 2. 3u 3. 3u + 2v

Unit vectors

• Unit vectors are vectors that have a magnitude of 1• The horizontal unit vector has

component form • The vertical unit vector has

component form

To find the unit vector of any non-vertical or non-horizontal vector:1. Find the magnitude of the vector2. Multiply the vector by the

reciprocal of its magnitude (basically divide the vector by its magnitude to give it a length of 1)

3. Perform the scalar multiplication on the appropriate form of the vector (the form the problem was written in)

Examples

• Determine the unit vector that has the same direction as each of the following:

1. 2. 3.

Example

• Write the vector with the given magnitude in the same direction as the vector given:

= 5

Assignment #2

6.3 Exercises#13-22, 25-28,35-40, 45-46

Examples