ANALYTIC GEOMETRY
In order to represent points in space, we first choose a fixed point O (the origin)
and three directed lines through O that are mutually perpendicular. These lines are
called the coordinate axes and are labeled 𝑥 − 𝑎𝑥𝑖𝑠, 𝑦 − 𝑎𝑥𝑖𝑠, and 𝑧 − 𝑎𝑥𝑖𝑠.
Usually, we think of the 𝑥 − and 𝑦 − 𝑎𝑥𝑖𝑠 as being horizontal and 𝑧 − 𝑎𝑥𝑖𝑠 being
vertical.
In three dimensional geometry, an equation in x, y and z represents a surface in 𝑅3 .
Example
Describe and sketch the surface in 𝑅3 represented by the equation
(a) 𝑧 = 3
(b) 𝑦 = 𝑥
Proposition
The distance |𝑃1 𝑃2 | between the points 𝑃1 (𝑥1 , 𝑥2 , 𝑥3 ) and 𝑃2 (𝑦1 , 𝑦2 , 𝑦3 ) is
|𝑃1 𝑃2 | = √(𝑦1 − 𝑥1 )2 + (𝑦2 − 𝑥2 )2 + (𝑦3 − 𝑥3 )2
VECTORS
Definition
⃗ = [𝑎1 , 𝑎2 ] of real numbers. A three
A two-dimensional vector is an ordered pair 𝒂
⃗ = [𝑎1 , 𝑎2 , 𝑎3 ] of real numbers. The
dimensional vector is an ordered triple 𝒂
⃗.
numbers 𝑎1 , 𝑎2 and 𝑎3 are called the components of the vector 𝒂
Definition
⃗ = [𝑎1 , 𝑎2 ] is given by
a) The length of a two dimensional vector 𝒂
⃗ | = √𝑎12 + 𝑎22 .
|𝒂
b) The length of a three dimensional vector ⃗𝒂= [𝑎1 , 𝑎2 , 𝑎3 ] is given by
⃗ | = √𝑎12 + 𝑎22 + 𝑎32 .
|𝒂
Definition
If ⃗𝒂 = [𝑎1 , 𝑎2 ] and ⃗𝒃 = [𝑏1 , 𝑏2 ] then the vector ⃗𝒂 + ⃗𝒃 is defined by
⃗ = [𝑎1 + 𝑏1 , 𝑎2 + 𝑏2 ].
⃗ +𝒃
𝒂
Similarly, for three dimensional vectors
⃗ + ⃗𝒃 = [𝑎1 + 𝑏1 , 𝑎2 + 𝑏2 , 𝑎3 + 𝑏3 ]
𝒂
Definition
⃗ = [𝑎1 , 𝑎2 ] and c is a scalar; then the vector c𝒂
⃗ is defined by c𝒂
⃗ = [𝑐𝑎1 , 𝑐𝑎2 ].
If 𝒂
⃗ = [𝑐𝑎1 , 𝑐𝑎2 , c𝑎3 ].
Similarly, for three dimensional vectors, c𝒂
There are three vectors in R3 that play a special role. Let 𝒊 = [1,0,0], 𝒋 = [0,1,0],
⃗ = [0,0,1]. Then 𝒊, 𝒋⃗ and 𝒌
⃗ are vectors that have length 1 and point in the
𝒌
direction of the positive 𝑥 −, 𝑦 −, and 𝑧 − 𝑎𝑥𝑖𝑠.
⃗ = [𝑎1 , 𝑎2 , 𝑎3 ] then we can write 𝒂
⃗ = [𝑎1 , 𝑎2 , 𝑎3 ] = [𝑎1 , 0, 0] + [0, 𝑎2 , 0] +
If 𝒂
⃗
[0, 0, 𝑎3 ] = 𝑎1 𝒊 + 𝑎2 𝒋 + 𝑎3 𝒌
⃗ are all unit
A unit vector is a vector whose length is 1. For instance 𝒊, 𝒋, and 𝒌
⃗ , then the unit vector that has the same direction as 𝒂
⃗ is not equal 𝟎
⃗ is
vectors. If 𝒂
⃗𝒃 = 𝒂⃗ .
⃗|
|𝒂
The dot product
Definition
⃗ = [𝑎1 , 𝑎2 , 𝑎3 ] and ⃗𝒃 = [𝑏1 , 𝑏2 , 𝑏3 ], then the dot product of 𝒂
⃗ and ⃗𝒃 is the
If 𝒂
⃗ ∗ ⃗𝒃 given by
number 𝒂
⃗ ∗ ⃗𝒃 = 𝑎1 𝑏1 + 𝑎2 𝑏2 + 𝑎3 𝑏3 .
𝒂
Theorem
⃗ , ⃗𝒃 and 𝒄
⃗ are vectors in the three dimensional space, and c a scalar, then
If 𝒂
⃗ ∗𝒂
⃗ = |𝒂|𝟐
1. 𝒂
⃗ ∗ ⃗𝒃= 𝒃 ∗ 𝒂
⃗
2. 𝒂
⃗ +𝒄
⃗ ∗ (𝒃
⃗)= 𝒂
⃗ ∗ ⃗𝒃 + 𝒂
⃗ ∗𝒄
⃗
3. 𝒂
⃗ )*b
⃗ ) ∗ ⃗𝒃 = 𝑐(𝒂
⃗ ∗ ⃗𝒃) = 𝒂
⃗ ∗ (𝑐𝒃
4. (c𝒂
⃗ =0
5. ⃗𝟎 ∗ 𝒂
Theorem
⃗ | cos 𝛼.
⃗ and ⃗𝒃, then 𝒂
⃗ ∗ ⃗𝒃 = |𝒂
⃗ | ∙ |𝒃
If α is the angle between the nonzero vectors 𝒂
Corollary
⃗ and ⃗𝒃, then cos 𝛼 =
If α is the angle between the nonzero vectors 𝒂
⃗
⃗ ∗𝒃
𝒂
⃗|
|𝒂
⃗ |∙|𝒃
Definition
𝜋
Two vectors are perpendicular or orthogonal if the angle between them is .
2
Theorem
⃗ and ⃗𝒃 are orthogonal if and only if 𝒂
⃗ ∗ ⃗𝒃 = 0.
Two vectors 𝒂
The cross product
Definition
⃗ = [𝑎1 , 𝑎2 , 𝑎3 ] and ⃗𝒃 = [𝑏1 , 𝑏2 , 𝑏3 ], then the cross product of 𝒂
⃗ and ⃗𝒃 is the
If 𝒂
vector
⃗
⃗ × ⃗𝒃 = (𝑎2 𝑏3 − 𝑎3 𝑏2 )𝒊 − (𝑎1 𝑏3 − 𝑎3 𝑏1 )𝒋 + (𝑎1 𝑏2 − 𝑎2 𝑏1 )𝒌
𝒂
Theorem
⃗ × ⃗𝒃 is orthogonal to both 𝒂
⃗ and ⃗𝒃.
The vector 𝒂
Theorem
⃗ | sin 𝛼.
⃗ and ⃗𝒃 (so 0 ≤ α ≤ π) then |𝒂
⃗ × ⃗𝒃| = |𝒂
⃗ | |𝒃
If α is the angle between 𝒂
Theorem
⃗ are parallel if and only if 𝒂
⃗ =𝟎
⃗.
⃗ and 𝒃
⃗ ×𝒃
Two nonzero vectors 𝒂
Theorem
⃗ and 𝒄
⃗ ,𝒃
⃗ are vectors and c a scalar, then
If 𝒂
⃗ ×𝒂
⃗ × ⃗𝒃 = −𝒃
⃗
1. 𝒂
⃗)
⃗ ) × ⃗𝒃 = 𝑐(𝒂
⃗ × ⃗𝒃) = 𝒂
⃗ × (𝑐𝒃
2. (𝑐𝒂
⃗ +𝒄
⃗ × (𝒃
⃗)=𝒂
⃗ × ⃗𝒃 + 𝒂
⃗ ×𝒄
⃗
3. 𝒂
⃗ + ⃗𝒃) × 𝒄
⃗ =𝒂
⃗ ×𝒄
⃗ + ⃗𝒃 × 𝒄
⃗
4. (𝒂
⃗ ×𝒄
⃗ × (𝒃
⃗ ) = (𝒂
⃗ × ⃗𝒃) × 𝒄
⃗.
5. 𝒂
Lines
A line L in three-dimensional space is determine by a point 𝑃0 (𝑥0 , 𝑦0 , 𝑧0 ) on the
⃗⃗⃗⃗𝟎 be the position vector of 𝑃0 (𝑥0 , 𝑦0 , 𝑧0 ), then every
line and the its direction. Let 𝒓
⃗⃗⃗⃗𝟎 + 𝑡𝒗
⃗ for some t, where 𝒗
⃗ is the direction of L.
point on L can be expressed as 𝒓
Therefore the equation
⃗ =𝒓
⃗⃗⃗⃗𝟎 + 𝑡𝒗
⃗
(1) 𝒓
⃗ = [𝑎, 𝑏, 𝑐], then t𝒗
⃗ = [𝑡𝑎, 𝑡𝑏, 𝑡𝑐].
is a vector equation of L. If the direction vector 𝒗
⃗ = [𝑥, 𝑦, 𝑧] and ⃗⃗⃗⃗
We can also write 𝒓
𝒓0 = [𝑥0 , 𝑦0 , 𝑧0 ], then the vector equation (1)
becomes
[𝑥, 𝑦, 𝑧] = [𝑥0 + 𝑡𝑎, 𝑦0 + 𝑡𝑏, 𝑧0 + 𝑡𝑐].
Thus
𝑥 = 𝑥0 + 𝑎𝑡
(2) {𝑦 = 𝑦0 + 𝑏𝑡.
𝑧 = 𝑧0 + 𝑐𝑡
These are called parametric equations of the line L. Another way of writing the
equation of a line L is to eliminate the parameter t from equations (2). If none of a,
b or c is zero, we can obtain
𝑥−𝑥0
𝑦−𝑦0
𝑧−𝑧0
(3)
=
=
.
𝑎
𝑏
𝑐
These equations are called the symmetric equations of L.
Planes
⃗ that is
A plane is determined by a point 𝑃0 (𝑥0 , 𝑦0 , 𝑧0 ) in the plane and a vector 𝒏
orthogonal to the plane. This orthogonal vector is called a normal vector. Let
⃗ = [𝑎, 𝑏, 𝑐]. Then
𝑃(𝑥, 𝑦, 𝑧) be a point on the plane and 𝒏
(4) 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑,
where 𝑑 = 𝑎𝑥0 + 𝑏𝑦0 + 𝑐𝑧0 is called the scalar equation of the plane through
⃗ = [𝑎, 𝑏, 𝑐]. Conversely, if a, b, and c are not
𝑃0 (𝑥0 , 𝑦0 , 𝑧0 ) with normal vector 𝒏
all zero, then equation (4) represents a plane with normal vector [𝑎, 𝑏, 𝑐].
Two planes are parallel if their normal vectors are parallel. If two planes are not
parallel, then they intersect in a straight line and the angle between the two planes
is defined as the acute angle between the two normal vectors.
The distance from the point (𝑥1 , 𝑦1 , 𝑧1 ) to the plane 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑 is
|𝑎𝑥1 +𝑏𝑦1 +𝑐𝑧1 −𝑑|
.
2
2
2
√𝑎 +𝑏 +𝑐