Intro to Vectors



Introduction to Vectors

Warm Up:

1) Convert (-4, -7) to polar form rounding [pic]to the nearest tenth.

2) Convert [2, [pic]] to rectangular form using exact values.

Vector :

The polar representation of a two-dimensional vector [pic] with non-negative magnitude r and direction [pic]measured from the polar axis is [r, [pic]].

- A vector is in standard form if it’s initial point is the pole.

The component representation of a plane vector [pic] is the ordered pair ([pic]), the rectangular coordinates of the endpoint of the vector in standard position.

- A vector is in standard position when it’s initial point is the origin.

Example1: A ship’s velocity is represented by [12, 82[pic]], where the first number is measured in miles per hour and the second is the number of degrees north of east that the ship travels.

a. Draw the arrow in standard position representing the velocity.

b. Describe the ship’s movement each hour in terms of a number east and a number of miles north.

Example 2: The arrow from (-1, 2) to (3, 5) represents a plane vector [pic].

a. Find the magnitude(length) and

direction of [pic].

b. Draw the standard position arrow for [pic].

Example 3: A vector [pic] represents a force of 5 pounds that is being exerted at an angle of [pic]with a positive x-axis. Find the x and y components of [pic].

The length or magnitude of the vector is also called the norm and is denoted by [pic]. If the

component representation of vector [pic]is (-4, 12) find [pic]to the nearest tenth.

Sum and Difference of Vectors

If [pic], then the sum

Example 1: If [pic] and [pic] find [pic]. (convert to component representation, then apply definition)

The vector [pic], the opposite of [pic], is the vector with the same magnitude as [pic] and

direction opposite to [pic].

Example 2: If [pic] find [pic] and graph both.

Convert [pic] to component form then find [pic] and graph both.

The difference [pic] is the vector [pic].

Example 3: Using [pic]from Example 1 above, find [pic].

Given: [pic]and [pic]

a) Graph and label both vectors.

b) Find [pic] and [pic].

c) Find [pic] then graph and label.

d) Draw a vector from the

endpoint of [pic] to the

endpoint of [pic], then

find its magnitude.

e)What else do you think is true?

f) What is [pic] called?

1) If [pic] and [pic] find the components of the resultant force [pic].

2) Two children push a friend on a sled with

the forces shown at the right.

a) Give the resultant force in polar form.

b) Interpret your answer to part a.

c) Which child is exerting more forward force?

Parallel Vectors and Equations of Vectors

Two vectors with the same or opposite direction are called parallel. That is, the polar representation of parallel vectors differ by a multiple of 180[pic].

a) Graph the following parallel vectors [pic] and [pic].

b) Convert the vectors from polar form to component form (using exact values).

We would say that [pic] is a scalar multiple of [pic] because [pic].

Example 1: Vector [pic] has initial point (2, 5) and ending point (-3, 10). Sketch the vector [pic]= -2[pic] in standard position. (find u in standard position first)

Example 2: Show that (2, 5) and (18, 45) are parallel.

Consider the line through the point P = (-4, 5) that is parallel to [pic]. A vector equation for this line would be (x, y) = (-4, 5) + t(7, 2) where t is the scale factor and is a real number. This means that all of the x-values are of the form x = -4 + 7t and the y – values, y = 5 + 2t. These are called the parametric equations.

Example 3: a) Write the equation of the line through point P (-3, -1) that is parallel to

[pic].

b) Write the parametric equations for the line above.

Dot Product

The dot product of [pic],

denoted by [pic]is the real number [pic].

If [pic] and [pic]then

1) find [pic].

2) find the length of [pic].

3) find the length of [pic].

4) find [pic]

5) find [pic]

Angle between the Vectors - Suppose that [pic]is the measure of the angle between two nonzero vectors [pic]and [pic], then [pic] where [pic] are not parallel.

1) Find the angle between vectors [pic] and [pic].

Two non-zero vectors are perpendicular if and only if their not product is zero. Perpendicular vectors are also called orthogonal.

2) Find the angle between vectors [pic] and [pic].

Three-Dimensional Coordinates

Example 1: Use the grid to plot and label the following points:

A(3, 1, 5)

B(-1, 2, 1)

C(2, -2, -1)

D (0, 0, 2)

E (0, 4, 1)

Distance in 3 – Space

The distance of the point (x, y, z) from the origin is [pic].

The distance between P = [pic] and Q = [pic]is given by[pic]

Use example 1 above in order to find

a) the distance point A is from the origin.

b) the distance point C is from point D.

The sphere with center (a, b, c) and radius r has equation [pic]

Example 1: Verify that the equation [pic] describes a sphere.

(find it’s center and radius by completing the square)

Example 2: Find the center and radius of the sphere with the equation [pic].

Vectors in 3 – Space

Let [pic] then

▪

▪ [pic] = [pic]

Examples: Let [pic]and find the following:

1) [pic]

2) [pic]

3) [pic]

4) [pic]

5) Find the angle between [pic].

6) Are [pic] orthogonal?

7) Find a vector that is orthogonal to [pic]

Cross Product

The vector operation called the cross product computes orthogonal vectors.

The cross product of two vectors [pic] is

In the picture below, the vector that is the cross product, [pic], is perpendicular to [pic] and perpendicular to [pic].

1) Find [pic]for vectors [pic].

2) Verify that [pic] is perpendicular to both [pic].

Lines and Planes in 3 – Space

Equations for a Line in 3 – Space

Suppose that P= [pic] is a fixed point in 3 – space and that [pic] is a fixed nonzero vector. Then there is one and only one straight line l in space that passes through P and is parallel to [pic]

The equation for this vector is (x, y, z,) = (a, b, c) + t[pic].

Example: Let m be the line passing through the two points (-1, 2, 3) and (4, -1, 5).

Describe m with a vector equation.

Equation for a Plane: The set of points (x, y, z) in space that satisfy the equation

ax + by + cz = d is a plane perpendicular to vector [pic] where [pic]is a nonzero vector.

(now we need a point and a perpendicular vector)

Example: Find vector and rectangular coordinate equations for the plane M that passes through the point P = (2, 3, 4) and is perpendicular to vector [pic]

1) Consider the line containing (1, 5, -2) parallel to the vector [pic].

a) Find a vector equation for the line.

b) Find the parametric equations for the line above.

2) Let m be the line with the following equation: (x, y, z,) = (-2, 3, 2) + t(3, -2, -3)

a) Is the point (10, -5, -8) on the line m?

b) Find an equation for the line m’ that is parallel to m and contains

the point (1, 2, 3)

3) Find an equation for the plane that is perpendicular to [pic]and contains the point (0, 5, 3).

-----------------------

initial point

end point

Pole

[pic]

[pic]

[pic]

Polar axis

standard position of

[pic]

E

W

S

N

y

x

[pic]

y

x

x

y

N

S

W

E

x

-z

z

-y

y

-x

[pic]

[pic]

P

z

y

x

y

x

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