Vectors

Vocabulary

100

A representation of a quantity that involves a magnitude and direction

Directed Line Segment (Vector)

200

Position where a Vector ends

Terminal point

300

The directed line segment drawn after completing u + v

Resultant

400

The name of a 3 dimensional circle

Sphere

500

The name of the product of 2 3D vectors that results in another vector

Cross Product

Vector Basics

100

Given u = <5, -1> and v = <-6,7> find:

2u - 3v

<28, -23>

200

Find the vector in component form that has an initial point A (-8, 4, 7) and terminal point B (1, -1, 9)

<9,-5, 2>

300

Find the magnitude of the vector <-7, 4>.  Write as an exact value.

Find the direction. Round to the nearest degree.

root 65

150 degrees

400

Write the exact position for the vector with a reference angle of 60 degrees in quadrant 3 and a magnitude of 18.  

<-9, -9 root 3>

500

Simplify:  2u  dot -3v

Given: 

u = <-root 8, 2root 5> & v=<5 root 2, 4 root 5>

-120

More with vectors

100

What value of b makes these vectors orthogonal?

<7, -4, 2> and <b, 6, 8> ?

8/7

200

Find the angle between vectors a and b.

a= <1, 2, 3> & b = <-3, 5, -9>

Round to the closest degree!

120^o

300

Find a vector orthogonal to both u and v.

u= <1, 1, 0>  and v = <-3, 4, 7>

Write as a linear combination.

7i - 7j + 7k            OR           -7i +7j -7k

400

Write the sum of u and v in magnitude and directional form.  Round to tenths.

____< cos ____,  sin ____>

u=5<cos 24^o, sin 24^o>

v=-9<cos 228^o, sin 228^o>

13.7<cos 39.5^o, sin 39.5^o>

500

Find the area of a triangle in 3D with the vertices:

A(0, 0, 0)  B(4, 8, -2) and C (-4, -3, 1)

Round to the nearest 100th!

10.25 square units

Applications

100

Write the sphere in standard form, identify the center (and name the plane the center lies in) and the radius as an exact value.

x² + y² + z² - 8x + 20y - 9 = 0

(x-4)²+(y+10)²+z²=125

Center: (4, -10, 0)  x-y plane

Radius: 5 root 5

200

Find the angle between the two vectors.

6(cos 93^o, sin 93^o)  and  10(cos 176^o, sin 176^o)

83^o

300

Find the projection of u onto v and write as the sum of two orthogonal vectors.

u = <3, 5>   and v = <-4, 4>

w₁ = <-1, 1> and w₂ = <4, 4>

<-1, 1> + <4, 4> = <3, 5>

400

Write the equation of a sphere with endpoints of the diameter at (-1, 4, 8) and (7, -2, 2). 

(x-3)² +(y-1)² + (z-5)² = 34

500

Find the volume of a parallelepiped that emanates from the origin and contains the following points as the terminal point of each side of the figure.

A(3, -4, 5)   B(-6, 1, 1)  C (12, 12, 12)

756 square units

More!

100

State the octant that contains the point] (1, -3, -5).

8

200

State all octants where xy > 0.

1, 3, 5, 7

300

Find the following:   W x V

given:  W = 2i - 7k and V = -3j + 6k

-21i - 12j - 6k

400

Make an equivalent statement given a, b and c are vectors.

a dot 2(b x c)

(ax2b) dot c  OR 

(axb) dot 2c OR

(2axb) dot c

500

Find a vector in the opposite direction of <-3, 8> with a magnitude of 5.

Round the the closest 10ths.

<1.8, - 4.7>