Transcript Advanced Precalculus Notes 8.6 Vectors in Space
Advanced Precalculus Notes 8.6 Vectors in Space
Distance Formula in Space:
d
x
2
x
1
y
2
y
1
z
2
z
1 2 Graph the point A(-1, 3, 2) and B(4, -2, 5) Find the distance between point A and B.
Find the Position Vector of the vector
v
P
1
P
2 if P 1 = (-1, 2, 3) and P 2 = (4, 6, 2)
If v = 2i +3j – 2k and w = 3i -4j + 5k find: a) v + w b) v – w c) 3v d) 2v - 3w
e) ||v||
Find a unit vector in the same direction as v = 2i -3j -6k
Find the following dot products: v = 2i -3j +6k a) v ∙ w w = 5i + 3j - k b) w ∙ v c) v ∙ v d) ||v||
Angle between Vectors: cos ||
u u
|| ||
v v
|| Find the angle between u = 2i -3j +6k v = 2i + 5j – k
Direction Angles of a Vector: given vector v = ai +bj +ck cos
a
||
v
|| angle between v and the unit vector i (the positive x-axis) cos
b
||
v
|| angle between v and the unit vector j (the positive y-axis) cos
c
||
v
|| angle between v and the unit vector k (the positive z-axis) Find the direction angles of v = - 3i +2j – 6k
cos 2 cos 2 cos 2 1 3 3 y-axis, and an acute angle with the positive z-axis.
V
v
[(cos )
i
(cos )
j
(cos )
k
]